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Re: just a question...

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  • Robin Garcia
    In other words, it is proven that there are infinetily many primes of the form n^2+m^2 but not for a determined n or m [Non-text portions of this message have
    Message 1 of 20 , Apr 13 9:07 AM
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      In other words, it is proven that there are infinetily many primes of the form n^2+m^2 but not for a determined n or m





      [Non-text portions of this message have been removed]
    • Chris Caldwell
      Robin Garcia: In other words, it is proven that there are infinetily many primes of the form n^2+m^2 but not for a determined n or m Yes. When you fix
      Message 2 of 20 , Apr 15 4:54 AM
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        Robin Garcia: In other words, it is proven that there are infinetily
        many primes of the form n^2+m^2 but not for a determined n or m

        Yes. When you fix either n or m, then you make the set a lot thinner
        (a much smaller set). The smaller the set, the harder it is to show it
        contains infinitely many primes. For example, it is easy to show
        n^2+m^2 has infinitely many primes, n^2+m^4 is much harder, and n^2+1 is
        currently beyond our reach.

        Chris.

        (Here "smaller" is often measured in the sense of natural densities,
        what percentage of them have this form less than x as x gets large.
        They are all countably infinite, so have a one-to-one correspondences.)
      • Ali Adams
        Greetings all,   Can someone please help with validating if the following 62-digit number is a prime or not:  
        Message 3 of 20 , Apr 21 8:01 PM
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          Greetings all,
           
          Can someone please help with validating if the following 62-digit number is a prime or not:
           
          11108195956680805165653650502135350605769090617575464617311659
           
          I am aware of IsProbablePrime but need a definite primality test.
           
          If I have to run the primality test on my machine then how do I get a rough estimate of the time it would take on say 2.33GHz QuadCore Intel CPU with 4Gb RAM? Even to the nearest month would be useful to know before I start the run.
           
          Thank you all,
           
          Ali
          www.heliwave.com




          [Non-text portions of this message have been removed]
        • maximilian_hasler
          using PARI/gp, ? ?isprime isprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0) if not. If flag is 0 or omitted, use a combination of
          Message 4 of 20 , Apr 22 5:49 AM
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            using PARI/gp,

            ? ?isprime
            isprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0) if not. If flag is 0 or omitted, use a combination of
            algorithms. If flag is 1, the primality is certified by the Pocklington-Lehmer Test. If flag is 2, the primality is certified
            using the APRCL test.

            ? isprime(11108195956680805165653650502135350605769090617575464617311659,1)
            %11 =
            [2 2 1]

            [37 2 1]

            [47 2 1]

            [283 2 1]

            [79539191 2 1]

            [134903310001 2 1]

            ?


            M.

            --- In primenumbers@yahoogroups.com, Ali Adams <alipoland@...> wrote:
            >
            > Greetings all,
            > �
            > Can someone please help with validating if the following 62-digit number is a prime or not:
            > �
            > 11108195956680805165653650502135350605769090617575464617311659
            > �
            > I am aware of IsProbablePrime but need a definite primality test.
            > �
            > If I have to run the primality test on my machine then how do I get a rough estimate�of the time it would take on�say 2.33GHz QuadCore Intel CPU with 4Gb RAM? Even to the nearest month would be useful to know before I start the run.
            > �
            > Thank you all,
            > �
            > Ali
            > www.heliwave.com
            >
            >
            >
            >
            > [Non-text portions of this message have been removed]
            >
          • Walter Nissen
            Greetings , UBASIC , running the supplied primality test , APRT-CLE , on an 11-year-old Pentium II at 266-MHz , produced this output :
            Message 5 of 20 , Apr 22 7:44 PM
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              Greetings ,

              UBASIC , running the supplied primality test , APRT-CLE , on
              an 11-year-old Pentium II at 266-MHz , produced this output :

              11108195956680805165653650502135350605769090617575464617311659 is prime.
              0:00:04

              4 seconds on the wall clock .
              Without experience running these programs , it is difficult even
              to ballpark run times .
              As you become more experienced , it will become easier .
              It would help if someone would collect experience from people
              and organize it .
              Even isolated cases would be somewhat helpful .
              I don't know if there is a textbook that gives recent info .
              Some web sites have some data , but how can they be located ?

              Cheers ,

              Walter
              http://upforthecount.com
            • Ali Adams
              Thanks Walter for your help. What about this 309-digit number?
              Message 6 of 20 , Apr 22 8:47 PM
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                Thanks Walter for your help.

                What about this 309-digit number?

                259336006801222696014182798990654577020329185417451253947966996203374778056958592994128470622708352120230964321433370545343196089458222530238878359827583627468563462233210398589090852507947007265127498998595582306765369537411175275870858814651979141558307396316154291312142703185675301452916463755740936626397

                How do I estimate how long it would take?
                 
                And even if after few days/weeks the result is prime.
                What is the meaning of the certificate to be validated by other programs mean?
                 
                Is there still a chance that it is not 100% prime?
                 
                Is the Elliptic Curve Method approximate?
                 
                Sorry for so many questions and thanks in advance.
                 
                Ali Adams
                God > infinity
                www.heliwave.com




                ________________________________
                From: Walter Nissen <nissen@...>
                To: Ali Adams <alipoland@...>; primenumbers@yahoogroups.com
                Cc: nissen@...
                Sent: Fri, April 23, 2010 10:44:14 AM
                Subject: Re: [PrimeNumbers] 62-digit IsPrime

                Greetings ,

                UBASIC , running the supplied primality test , APRT-CLE , on
                an 11-year-old Pentium II at 266-MHz , produced this output :

                11108195956680805165653650502135350605769090617575464617311659 is prime.
                  0:00:04

                4 seconds on the wall clock .
                Without experience running these programs , it is difficult even
                to ballpark run times .
                As you become more experienced , it will become easier .
                It would help if someone would collect experience from people
                and organize it .
                Even isolated cases would be somewhat helpful .
                I don't know if there is a textbook that gives recent info .
                Some web sites have some data , but how can they be located ?

                Cheers ,

                Walter
                http://upforthecount.com




                [Non-text portions of this message have been removed]
              • Andy Steward
                Hi All, ... I also use that program for numbers below 250 digits. I have modified it to run in batch mode: an input file has the serial number (from my
                Message 7 of 20 , Apr 22 10:49 PM
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                  Hi All,

                  Walter Nissen wrote:
                  >UBASIC , running the supplied primality test , APRT-CLE , on
                  >an 11-year-old Pentium II at 266-MHz , produced this output :

                  I also use that program for numbers below 250 digits. I have modified
                  it to run in batch mode: an input file has the serial number (from my
                  database) of the PrP and the PrP itself, one per line. The output file
                  has the same serial number followed by "P" for prime, "C" for composite
                  or "U" for undetermined (this last has never happened). If anyone is
                  interested, I will happily email them the code.

                  >Without experience running these programs, it is difficult even
                  >to ballpark run times.

                  I have over 100,000 data points for the Ubasic code but they are of no
                  use for statistics as I do not record which processor did the run and
                  the code does not usually run with exclusive use of the CPU.

                  For larger PrPs, I (and almost everyone else) use Primo, available from
                  http://www.ellipsa.eu/public/primo/primo.html

                  I have collected stats on around 2600 runs of Primo. For estimating
                  purposes I use the data from the 64 largest primes I have proved with
                  the latest major release (all 3.x versions). Currently:
                  Let K = Kilo Digits = (Base 10 log of input number) / 1000
                  Let G = GigaHertz * Days
                  Then G = 0.0346 * (K ^ 4.49)

                  Those stats are based on inputs from about 3000 to 7400 digits. For small
                  inputs, this estimate (0.01 GHz seconds for the 62 digit input) is swamped
                  by overhead.

                  HTH,
                  Andy
                • maximilian_hasler
                  It can t be prime because it s not pseudoprime. I suggest you install PARI/gp (done in less than a minute) and do such preliminary checks prior to posting in
                  Message 8 of 20 , Apr 24 5:16 AM
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                    It can't be prime because it's not pseudoprime.
                    I suggest you install PARI/gp (done in less than a minute)
                    and do such preliminary checks prior to posting in this group.

                    Maximilian

                    --- In primenumbers@yahoogroups.com, Ali Adams <alipoland@...> wrote:
                    >
                    > Thanks Walter�for your help.
                    >
                    > What about this 309-digit number?
                    >
                    > 259336006801222696014182798990654577020329185417451253947966996203374778056958592994128470622708352120230964321433370545343196089458222530238878359827583627468563462233210398589090852507947007265127498998595582306765369537411175275870858814651979141558307396316154291312142703185675301452916463755740936626397
                    >
                    > How do I estimate how long it would take?
                    > �
                    > And even if after few days/weeks the result is prime.
                    > What is the meaning of the certificate to be validated by other programs mean?
                    > �
                    > Is there still a chance that it is not 100% prime?
                    > �
                    > Is the Elliptic Curve Method approximate?
                    > �
                    > Sorry for so many questions and thanks in advance.
                    > �
                    > Ali Adams
                    > God > infinity
                    > www.heliwave.com
                    >
                    >
                    >
                    >
                    > ________________________________
                    > From: Walter Nissen <nissen@...>
                    > To: Ali Adams <alipoland@...>; primenumbers@yahoogroups.com
                    > Cc: nissen@...
                    > Sent: Fri, April 23, 2010 10:44:14 AM
                    > Subject: Re: [PrimeNumbers] 62-digit IsPrime
                    >
                    > Greetings ,
                    >
                    > UBASIC , running the supplied primality test , APRT-CLE , on
                    > an 11-year-old Pentium II at 266-MHz , produced this output :
                    >
                    > 11108195956680805165653650502135350605769090617575464617311659 is prime.
                    > � 0:00:04
                    >
                    > 4 seconds on the wall clock .
                    > Without experience running these programs , it is difficult even
                    > to ballpark run times .
                    > As you become more experienced , it will become easier .
                    > It would help if someone would collect experience from people
                    > and organize it .
                    > Even isolated cases would be somewhat helpful .
                    > I don't know if there is a textbook that gives recent info .
                    > Some web sites have some data , but how can they be located ?
                    >
                    > Cheers ,
                    >
                    > Walter
                    > http://upforthecount.com
                    >
                    >
                    >
                    >
                    > [Non-text portions of this message have been removed]
                    >
                  • Norman Luhn
                    The next prime number is your number+334 N=...626731 PRIMO takes 6,25s to create a certificate. Best Norman [Non-text portions of this message have been
                    Message 9 of 20 , Apr 24 5:28 AM
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                      The next prime number is your number+334
                      N=...626731


                      PRIMO takes 6,25s to create a certificate.

                      Best

                      Norman














                      [Non-text portions of this message have been removed]
                    • Alan Eliasen
                      ... (in another posting, asked) ... (Which is not even probable-prime, as one single strong pseudoprime test shows.) I m about to sermonize, so be prepared, my
                      Message 10 of 20 , Apr 25 12:59 AM
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                        On 04/21/2010 09:01 PM, Ali Adams wrote:
                        > Greetings all,
                        >
                        > Can someone please help with validating if the following 62-digit
                        > number is a prime or not:
                        >
                        > 11108195956680805165653650502135350605769090617575464617311659
                        >
                        > I am aware of IsProbablePrime but need a definite primality test.

                        (in another posting, asked)
                        > What about this 309-digit number?

                        >259336006801222696014182798990654577020329185417451253947966996203374778056958592994128470622708352120230964321433370545343196089458222530238878359827583627468563462233210398589090852507947007265127498998595582306765369537411175275870858814651979141558307396316154291312142703185675301452916463755740936626397

                        (Which is not even probable-prime, as one single strong pseudoprime
                        test shows.)

                        I'm about to sermonize, so be prepared, my brothers and sisters.

                        Whenever someone says they "need" a definite primality test, they, by
                        their actions, usually prove that they don't understand that a
                        primality-proving test, by itself, is not an infallible proof of
                        primality (in the real world) and they rarely if ever do the correct
                        thing to decrease the probability of error, giving them a result that's
                        effectively no more certain than a working pseudoprime test (and, by
                        probabilities and failure modes that I'll cite here, probably quite a
                        bit weaker. And by "quite a bit" I mean 20 orders of magnitude, easy.)

                        Note: Keep in mind that all of this analysis applies in the *real
                        world*, not to some perfect, fictitious mathematical abstraction where
                        programmers don't make errors and hardware never fails and you can
                        ignore most algorithms and divide out big pesky constants because the
                        ideal theoretical asymptotic performance is always what you get when you
                        run programs.

                        It is simple to implement and perform a strong pseudoprime test in
                        which the probability that a randomly-chosen composite number is
                        mistakenly stated to be prime is so low that it would never happen in
                        your lifetime.[1] Not only that, you can trivially make the probability
                        of this mistake so low that you could test trillions of numbers every
                        second for the lifetime of the universe, and the probability of *any* of
                        those tests failing are still astronomically low, and that's even taking
                        the old ultra-conservative bound that as many as 1/4 of strong
                        pseudoprime tests can fail. See citations in the link below for numbers
                        about how conservative that estimate really is.

                        Many people have pointed out over the years that the probability that
                        your *hardware* or *software* fails (say, due to a high-energy particle
                        passing through your processor, or random thermal drift of electrons[2],
                        or a bad transistor) during your primality test or primality "proof"
                        becomes rapidly much, much higher than the probability that, say, a
                        Rabin-Miller test incorrectly declares a composite to be prime.
                        Depending on the size of the number, this is true *even if you only do
                        one pass of a Rabin-Miller test*, and the probability of error in the
                        algorithm is far less than the probability of hardware failure, possibly
                        by hundreds or thousands of tens of thousands of *orders of magnitude*!

                        It may also be far, far more probable that the number you meant to
                        test or the reply (with certificate) was corrupted during communication
                        with someone else. For example, TCP only has a 16-bit checksum, and can
                        miss, say, two single-bit errors separated by 16 bits, for a possible
                        undetected error rate of 2^-16, or 1/65536 in a noisy channel. (Other
                        error-correcting algorithms apply to communication across the internet,
                        and most channels have fairly good s/n ratios so the end-to-end
                        probability of error is luckily usually lower than this.)

                        There's also the probability that Yahoo does something stupid with
                        the long line and cuts off a digit or something. (Some would say the
                        probability of Yahoo's software doing something stupid when adding their
                        cruft to a posting is 1, but that's being mean. Their software isn't
                        really that bad. The probability is actually 1-epsilon.)

                        To see approximate probabilities of the Rabin-Miller test failing
                        for certain number sizes, even with a *single* round of tests, see:

                        http://primes.utm.edu/notes/prp_prob.html

                        Those probabilities of failure, especially for large numbers, are
                        mind-bogglingly small! Far smaller than the probability of some other
                        source of error.

                        Note that for the 309-digit number posted above, the probability of a
                        strong pseudoprime test mistakenly returning "prime" for a composite
                        number is less than 5.8*10^-29. Is that probability higher than other
                        probabilities we've already listed? (And note that since a strong
                        pseudoprime test easily catches that the number is actually composite.
                        No matter what base you choose. I'm sure that many here wouldn't
                        hesistate to give a cash reward for anyone who can find *any* prime base
                        smaller than the number for which a strong pseudoprime test fails. I
                        will initially offer all the money in my wallet. And I haven't even
                        tried to look for one. I'm that confident in the probabilities, or my
                        wallet is at its usual sad level.)

                        It's hard to approximate the probability that a particular piece of
                        software or hardware or communication will fail, but you can never
                        expect your primality "proof" to be any stronger than the most likely
                        error in the chain.

                        If you really *need* a prime number, you *must* then be sure to take
                        the certificate produced by one program's prime number proof *and verify
                        this certificate*, presumably with different software on a different
                        machine and hardware architecture! (I feel I should repeat this
                        sentence many times!) And then verify it again with another piece of
                        software! If you *don't* do this, the probability that the primality
                        "proof" is in error is approximately the probability of the thing most
                        likely to go wrong in the entire chain. (Again, maybe cosmic rays,
                        software bugs, Pentiums, race conditions in software, power glitches,
                        probability of human error, etc.) The primality certificate gives you a
                        way of verifying (without performing the whole proof again) that the
                        proof is indeed valid for that number.

                        However, there's nothing that prevents even the *verification* of the
                        certificate from having a similar unlikely failure! (Or similar
                        implementation bugs.)[3] Assuming the implementations and hardware are
                        completely independent, the best you can do is to multiply the
                        probability of failure in both systems. Thus, if the probability of
                        failure in each system independently is 10^-32, then the probability
                        that *both* fail completely independently but in a way that gives the
                        same flawed result is at best 10^-64. Look at the URL cited above and
                        compare that to the probability of a probable-prime test failing. Is it
                        larger or smaller? If the probability is larger, the likelihood of your
                        "proof" being better than a pseudoprime test is purely illusory.

                        I'm not sure what the exact failure probability for a single
                        instruction is in modern hardware, but it's almost certainly *not*
                        better than 10^-32. (Multiply this by the number of instructions
                        required to perform the calculation.)

                        One of the best sources of information I've seen about actual failure
                        rates in installed hardware was done by the SETI@Home team, (which had
                        the world's largest distributed supercomputer at some points) which
                        cited detected error rates in returned work units (each work unit was
                        given to at least two people for validation, so errors could be
                        detected.) I don't remember the exact numbers, but the actual failure
                        rate appeared to be many, many orders of magnitude lower than 10^-32. I
                        don't seem to be able to find these stats at the moment. Does anyone
                        have the link?[4]

                        If you haven't verified a primality certificate independently to
                        decrease your probability of hardware or software error, (and there is
                        *always* a non-zero probability of error on real physical hardware and
                        where humans are involved, even in a primality "proof") you're likely
                        not doing any better than a probabilistic prime test, in which the
                        unreliability of hardware, software, communications, and humans, rapidly
                        become the limiting factors.

                        You're also much more likely that some random mischiefmaker will say,
                        "sure it's prime" when it's not because they're annoyed with you for not
                        doing these simple tests yourself, and for refusing to take advice on
                        tools that will do the work for you. But if people didn't refuse to
                        listen, I wouldn't get a chance to gratuitously sermonize.

                        If you didn't verify the primality certificate as many ways as
                        possible, you clearly can't claim you understand that you "need" a prime
                        number, and don't really understand the probability of all the different
                        ways that failure could have occurred. Do you even know that the
                        primality certificate that was posted here was valid for the number you
                        gave, or did someone maybe miss the last digit digit during
                        cut-and-paste? Do you even know that the certificate was anything but a
                        cat walking across a keyboard? The probability that *you*
                        cut-and-pasted the number incorrectly is orders of magnitude higher than
                        probability of failure of a pseudoprime algorithm.[5]

                        Homework: 1.) Can anyone else estimate probabilities of certain
                        types of errors? What do you think are the probabilities of various
                        failure modes in posting "prove this number prime for me" to a public
                        group? Do any of those probabilities exceed that of failure of a
                        probabilistic test? (Hint: The answer is yes.)

                        2.) Even if you get a provable prime number handed down impeachably
                        from the ghost of Fermat, what's the probability that when you're
                        *using* this prime number that something goes wrong in those
                        calculations? (Hint: Weakest link in the chain.)

                        ================

                        Footnotes:

                        [1] I intentionally state "randomly-chosen" because there are ways
                        to generate a very sparse set of numbers that can fool one of these
                        tests if you know in advance the bases it's going to test, which is why
                        most Rabin-Miller tests choose some bases randomly if there is the
                        potential for adversarial assault. See, for example:
                        François Arnault, Constructing Carmichael numbers which are strong
                        pseudoprimes to several bases, Journal of Symbolic Computation, v.20
                        n.2, p.151-161, Aug. 1995

                        [2] There's a certain nonzero probability that an electron will go
                        "the wrong way" and potentially tunnel "backwards" through a potential
                        barrier. As voltages used in processors get lower, and as gates get
                        smaller and the number of electrons required to switch a gate get fewer,
                        this becomes increasingly more probable. (I don't have my "Feynman
                        Lectures on Computation" at hand or I'd post some equations for this
                        probability. Very highly recommended!
                        http://tinyurl.com/9q4p8o )

                        [3] Multiple programmers creating similar software bugs is more
                        common than you might think. Sun's Java implementation had a famous bug
                        in their calculation of the Jacobi symbol, which caused
                        primality-testing routines to fail:

                        http://bugs.sun.com/bugdatabase/view_bug.do?bug_id=4624738

                        When I implemented a function to calculate the Jacobi symbol in my
                        Frink programming language ( http://futureboy.us/frinkdocs/ ) I
                        initially created a similar bug, because my algorithm didn't work for
                        some negative numbers either. I didn't use Sun's algorithm, but I can
                        see how they went wrong. The algorithm often cited on the web, in
                        several number theory books, etc. needs preconditioning to handle
                        negative numbers correctly. Even more insanely, when evaluating
                        negative numbers, Java uses different sign conventions for the %
                        (modulus) operator (used for "int" values) and the BigInteger.mod(n)
                        function! It's easy to see how Sun even confuses their own programmers.

                        This bug in Sun's implementation caused some significant pain. Their
                        primality testing was originally a probabilistic Rabin-Miller test (with
                        probabilities that can easily be set that it won't return a wrong result
                        during the lifetime of the universe, and many users set them *way*
                        higher than that to be very safe) yet that bug caused failures much more
                        often, and introduced a method of failure that Rabin-Miller *can't*
                        produce, (Rabin-Miller can't ever declare a prime to be composite, but
                        it can, very rarely, declare a composite to be probably prime,) which is
                        why this failure mode was particularly unexpected to a lot of people,
                        and caused numbers above a certain size to fail mysteriously and
                        sporadically.

                        Note: When I was informed of this bug in my implementation of my
                        JacobiSymbol[x] function, (it didn't affect my primality testing, which
                        has always worked properly,) I was ashamed that I wasn't able to release
                        a fix until I got home from work later that day. Sun, on the other
                        hand, took *3 years* to release a fix. (This bug was present in Java
                        versions 1.3 through 1.4.2.)

                        [4] Of course, some of their "failures" were due to intentional
                        attempts of people to corrupt their results, so exact probabilities of
                        failure are impossible to come by. There were interesting stories,
                        though, of *all* or *almost all* of the work units returned by several
                        well-meaning participants returning incorrect results. The SETI team
                        contacted these people and were able to verify that their computers'
                        floating-point units were indeed failing. (Sometimes subtly enough to
                        not make everything crash, but enough to make all extended calculations
                        wrong.) This is interesting and probably indicates that the frequency
                        of subtly or explicitly broken processors in the world is far less than
                        10^-9, setting a bound for what reliability we might expect for
                        primality testing.

                        [5] Don't laugh. This happened on this very list. A few years ago,
                        an intrepid researcher stated that they had been running a computer for
                        3+ years to find the factors of one of the RSA factoring challenge
                        numbers. Awesome persistence! And then one day it beeped! (I can't
                        imagine the excitement!) He announced to this list that he had
                        submitted his solution to RSA and was awaiting confirmation of the
                        factors, and he wasn't sure if the numbers were factors. I asked him
                        the obvious question, "did you multiply the two factors together and did
                        they come up with the original number?" The next day, with a leaden
                        heart, he responded to me and indicated that he had accidentally cut off
                        a digit when pasting in the original RSA number to his factoring
                        program. It still hurts me to think about it.

                        --
                        Alan Eliasen
                        eliasen@...
                        http://futureboy.us/
                      • Jens Kruse Andersen
                        ... The below is largely my own speculation not based on careful research. Your numbers apparently assume there are no hardware, software, operator or other
                        Message 11 of 20 , Apr 25 6:14 AM
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                          Alan Eliasen wrote:
                          > It is simple to implement and perform a strong pseudoprime test in
                          > which the probability that a randomly-chosen composite number is
                          > mistakenly stated to be prime is so low that it would never happen in
                          > your lifetime.[1] Not only that, you can trivially make the probability
                          > of this mistake so low that you could test trillions of numbers every
                          > second for the lifetime of the universe, and the probability of *any* of
                          > those tests failing are still astronomically low, and that's even taking
                          > the old ultra-conservative bound that as many as 1/4 of strong
                          > pseudoprime tests can fail. See citations in the link below for numbers
                          > about how conservative that estimate really is.

                          The below is largely my own speculation not based on careful research.

                          Your numbers apparently assume there are no hardware, software,
                          operator or other errors when the pseudoprime tests are performed.
                          But they can have the same type of errors as primality proofs.
                          In practice it seems more important to use independent reliable software
                          and hardware for the pseudoprime tests than to run a large number of them.
                          And if you both have pseudoprime tests saying composite and probable
                          prime on a large number then be extremely careful before claiming to have
                          proved compositeness.

                          Fast pseudoprime test programs may implement more complicated
                          algorithms with greater risk of programming error, and may be written
                          in assembler that tends to produce more errors for human programmers
                          (partly because of increassed source code size), so it may be better to
                          run a smaller number of pseudoprime tests with a slow simple program
                          than a large number with a fast program.

                          If you run the same type of pseudoprime test with multiple supposedly
                          independent programs then some things risk causing the same error,
                          for example:
                          An error in a text read by multiple programmers.
                          A systematic hardware error affecting operations likely to be performed
                          by different programs implementing the same algorithm.
                          A tricky part of the implementation which may fool multiple programmers.
                          An error in a software library or other routine used by multiple programs.

                          So it also seems a good idea in practice to use different types of
                          pseudoprime tests instead of many uses of the one with the best
                          theoretical accuracy.

                          If the same person makes many tests then a copy-and-paste error
                          or lack of honesty, knowledge, intelligence, carefulness or other
                          human factors (some of them may affect yourself without your
                          knowledge) can make all tests invalid, so also get multiple people
                          to run tests.

                          In brief, if you want to increase the "real" chance that a number is
                          prime then I think you should bet more on independence of as
                          much as possible (software, hardware, algorithms, people, source
                          of the number), than on increasing the number of pseudoprime tests.
                          And aim for not only two independent of something but as many as
                          possible or practical.

                          However, in practice the negative consequences of an alleged prime
                          being composite are usually so small that I don't think people should
                          bother with all my advice (I don't myself).
                          Most large primes are just announced with no practical consequence
                          (except maybe to the reputation of you or the used software) if
                          somebody else proves them composite.
                          It will often be more important to an audience of an announced prime
                          that you say "Trusted program X proved primality" than you argue
                          about the microscopic risk that something went wrong in all the
                          pseudoprime tests.

                          --
                          Jens Kruse Andersen
                        • Ali Adams
                          Thank  you for the reply Alan and advice taken well. Here in China I don t have access to the PRIMO website and the one I am using now is
                          Message 12 of 20 , Apr 25 6:22 AM
                          • 0 Attachment
                            Thank  you for the reply Alan and advice taken well.
                            Here in China I don't have access to the PRIMO website and the one I am using now is http://www.alpertron.com.ar/ECM.HTM which says composite but still no prime factors out yet.
                             
                            I am interested in either the number being a prime and then see if its digit sum is also prime (additve prime number). Alternatively if composite then the maths stop and the interpretation of the prime factors starts.
                             
                            Thank you again for your advice and I am a software engineer and I know the weakest link :)
                             
                            Salam,
                             
                            Ali
                            God > infinity




                            ________________________________
                            From: Alan Eliasen <eliasen@...>
                            To: Ali Adams <alipoland@...>; Prime Number <primenumbers@yahoogroups.com>
                            Sent: Sun, April 25, 2010 3:59:17 PM
                            Subject: [PrimeNumbers] Strong pseudosermon (was 62-digit IsPrime)

                             
                            On 04/21/2010 09:01 PM, Ali Adams wrote:
                            > Greetings all,
                            >
                            > Can someone please help with validating if the following 62-digit
                            > number is a prime or not:
                            >
                            > 1110819595668080516 5653650502135350 6057690906175754 64617311659
                            >
                            > I am aware of IsProbablePrime but need a definite primality test.

                            (in another posting, asked)
                            > What about this 309-digit number?

                            >259336006801222696 0141827989906545 7702032918541745 1253947966996203 3747780569585929 9412847062270835 2120230964321433 3705453431960894 5822253023887835 9827583627468563 4622332103985890 9085250794700726 5127498998595582 3067653695374111 7527587085881465 1979141558307396 3161542913121427 0318567530145291 6463755740936626 397

                            (Which is not even probable-prime, as one single strong pseudoprime
                            test shows.)

                            I'm about to sermonize, so be prepared, my brothers and sisters.

                            Whenever someone says they "need" a definite primality test, they, by
                            their actions, usually prove that they don't understand that a
                            primality-proving test, by itself, is not an infallible proof of
                            primality (in the real world) and they rarely if ever do the correct
                            thing to decrease the probability of error, giving them a result that's
                            effectively no more certain than a working pseudoprime test (and, by
                            probabilities and failure modes that I'll cite here, probably quite a
                            bit weaker. And by "quite a bit" I mean 20 orders of magnitude, easy.)

                            Note: Keep in mind that all of this analysis applies in the *real
                            world*, not to some perfect, fictitious mathematical abstraction where
                            programmers don't make errors and hardware never fails and you can
                            ignore most algorithms and divide out big pesky constants because the
                            ideal theoretical asymptotic performance is always what you get when you
                            run programs.

                            It is simple to implement and perform a strong pseudoprime test in
                            which the probability that a randomly-chosen composite number is
                            mistakenly stated to be prime is so low that it would never happen in
                            your lifetime.[1] Not only that, you can trivially make the probability
                            of this mistake so low that you could test trillions of numbers every
                            second for the lifetime of the universe, and the probability of *any* of
                            those tests failing are still astronomically low, and that's even taking
                            the old ultra-conservative bound that as many as 1/4 of strong
                            pseudoprime tests can fail. See citations in the link below for numbers
                            about how conservative that estimate really is.

                            Many people have pointed out over the years that the probability that
                            your *hardware* or *software* fails (say, due to a high-energy particle
                            passing through your processor, or random thermal drift of electrons[2] ,
                            or a bad transistor) during your primality test or primality "proof"
                            becomes rapidly much, much higher than the probability that, say, a
                            Rabin-Miller test incorrectly declares a composite to be prime.
                            Depending on the size of the number, this is true *even if you only do
                            one pass of a Rabin-Miller test*, and the probability of error in the
                            algorithm is far less than the probability of hardware failure, possibly
                            by hundreds or thousands of tens of thousands of *orders of magnitude*!

                            It may also be far, far more probable that the number you meant to
                            test or the reply (with certificate) was corrupted during communication
                            with someone else. For example, TCP only has a 16-bit checksum, and can
                            miss, say, two single-bit errors separated by 16 bits, for a possible
                            undetected error rate of 2^-16, or 1/65536 in a noisy channel. (Other
                            error-correcting algorithms apply to communication across the internet,
                            and most channels have fairly good s/n ratios so the end-to-end
                            probability of error is luckily usually lower than this.)

                            There's also the probability that Yahoo does something stupid with
                            the long line and cuts off a digit or something. (Some would say the
                            probability of Yahoo's software doing something stupid when adding their
                            cruft to a posting is 1, but that's being mean. Their software isn't
                            really that bad. The probability is actually 1-epsilon.)

                            To see approximate probabilities of the Rabin-Miller test failing
                            for certain number sizes, even with a *single* round of tests, see:

                            http://primes. utm.edu/notes/ prp_prob. html

                            Those probabilities of failure, especially for large numbers, are
                            mind-bogglingly small! Far smaller than the probability of some other
                            source of error.

                            Note that for the 309-digit number posted above, the probability of a
                            strong pseudoprime test mistakenly returning "prime" for a composite
                            number is less than 5.8*10^-29. Is that probability higher than other
                            probabilities we've already listed? (And note that since a strong
                            pseudoprime test easily catches that the number is actually composite.
                            No matter what base you choose. I'm sure that many here wouldn't
                            hesistate to give a cash reward for anyone who can find *any* prime base
                            smaller than the number for which a strong pseudoprime test fails. I
                            will initially offer all the money in my wallet. And I haven't even
                            tried to look for one. I'm that confident in the probabilities, or my
                            wallet is at its usual sad level.)

                            It's hard to approximate the probability that a particular piece of
                            software or hardware or communication will fail, but you can never
                            expect your primality "proof" to be any stronger than the most likely
                            error in the chain.

                            If you really *need* a prime number, you *must* then be sure to take
                            the certificate produced by one program's prime number proof *and verify
                            this certificate* , presumably with different software on a different
                            machine and hardware architecture! (I feel I should repeat this
                            sentence many times!) And then verify it again with another piece of
                            software! If you *don't* do this, the probability that the primality
                            "proof" is in error is approximately the probability of the thing most
                            likely to go wrong in the entire chain. (Again, maybe cosmic rays,
                            software bugs, Pentiums, race conditions in software, power glitches,
                            probability of human error, etc.) The primality certificate gives you a
                            way of verifying (without performing the whole proof again) that the
                            proof is indeed valid for that number.

                            However, there's nothing that prevents even the *verification* of the
                            certificate from having a similar unlikely failure! (Or similar
                            implementation bugs.)[3] Assuming the implementations and hardware are
                            completely independent, the best you can do is to multiply the
                            probability of failure in both systems. Thus, if the probability of
                            failure in each system independently is 10^-32, then the probability
                            that *both* fail completely independently but in a way that gives the
                            same flawed result is at best 10^-64. Look at the URL cited above and
                            compare that to the probability of a probable-prime test failing. Is it
                            larger or smaller? If the probability is larger, the likelihood of your
                            "proof" being better than a pseudoprime test is purely illusory.

                            I'm not sure what the exact failure probability for a single
                            instruction is in modern hardware, but it's almost certainly *not*
                            better than 10^-32. (Multiply this by the number of instructions
                            required to perform the calculation. )

                            One of the best sources of information I've seen about actual failure
                            rates in installed hardware was done by the SETI@Home team, (which had
                            the world's largest distributed supercomputer at some points) which
                            cited detected error rates in returned work units (each work unit was
                            given to at least two people for validation, so errors could be
                            detected.) I don't remember the exact numbers, but the actual failure
                            rate appeared to be many, many orders of magnitude lower than 10^-32. I
                            don't seem to be able to find these stats at the moment. Does anyone
                            have the link?[4]

                            If you haven't verified a primality certificate independently to
                            decrease your probability of hardware or software error, (and there is
                            *always* a non-zero probability of error on real physical hardware and
                            where humans are involved, even in a primality "proof") you're likely
                            not doing any better than a probabilistic prime test, in which the
                            unreliability of hardware, software, communications, and humans, rapidly
                            become the limiting factors.

                            You're also much more likely that some random mischiefmaker will say,
                            "sure it's prime" when it's not because they're annoyed with you for not
                            doing these simple tests yourself, and for refusing to take advice on
                            tools that will do the work for you. But if people didn't refuse to
                            listen, I wouldn't get a chance to gratuitously sermonize.

                            If you didn't verify the primality certificate as many ways as
                            possible, you clearly can't claim you understand that you "need" a prime
                            number, and don't really understand the probability of all the different
                            ways that failure could have occurred. Do you even know that the
                            primality certificate that was posted here was valid for the number you
                            gave, or did someone maybe miss the last digit digit during
                            cut-and-paste? Do you even know that the certificate was anything but a
                            cat walking across a keyboard? The probability that *you*
                            cut-and-pasted the number incorrectly is orders of magnitude higher than
                            probability of failure of a pseudoprime algorithm.[5]

                            Homework: 1.) Can anyone else estimate probabilities of certain
                            types of errors? What do you think are the probabilities of various
                            failure modes in posting "prove this number prime for me" to a public
                            group? Do any of those probabilities exceed that of failure of a
                            probabilistic test? (Hint: The answer is yes.)

                            2.) Even if you get a provable prime number handed down impeachably
                            from the ghost of Fermat, what's the probability that when you're
                            *using* this prime number that something goes wrong in those
                            calculations? (Hint: Weakest link in the chain.)

                            ============ ====

                            Footnotes:

                            [1] I intentionally state "randomly-chosen" because there are ways
                            to generate a very sparse set of numbers that can fool one of these
                            tests if you know in advance the bases it's going to test, which is why
                            most Rabin-Miller tests choose some bases randomly if there is the
                            potential for adversarial assault. See, for example:
                            François Arnault, Constructing Carmichael numbers which are strong
                            pseudoprimes to several bases, Journal of Symbolic Computation, v.20
                            n.2, p.151-161, Aug. 1995

                            [2] There's a certain nonzero probability that an electron will go
                            "the wrong way" and potentially tunnel "backwards" through a potential
                            barrier. As voltages used in processors get lower, and as gates get
                            smaller and the number of electrons required to switch a gate get fewer,
                            this becomes increasingly more probable. (I don't have my "Feynman
                            Lectures on Computation" at hand or I'd post some equations for this
                            probability. Very highly recommended!
                            http://tinyurl. com/9q4p8o )

                            [3] Multiple programmers creating similar software bugs is more
                            common than you might think. Sun's Java implementation had a famous bug
                            in their calculation of the Jacobi symbol, which caused
                            primality-testing routines to fail:

                            http://bugs. sun.com/bugdatab ase/view_ bug.do?bug_ id=4624738

                            When I implemented a function to calculate the Jacobi symbol in my
                            Frink programming language ( http://futureboy. us/frinkdocs/ ) I
                            initially created a similar bug, because my algorithm didn't work for
                            some negative numbers either. I didn't use Sun's algorithm, but I can
                            see how they went wrong. The algorithm often cited on the web, in
                            several number theory books, etc. needs preconditioning to handle
                            negative numbers correctly. Even more insanely, when evaluating
                            negative numbers, Java uses different sign conventions for the %
                            (modulus) operator (used for "int" values) and the BigInteger.mod( n)
                            function! It's easy to see how Sun even confuses their own programmers.

                            This bug in Sun's implementation caused some significant pain. Their
                            primality testing was originally a probabilistic Rabin-Miller test (with
                            probabilities that can easily be set that it won't return a wrong result
                            during the lifetime of the universe, and many users set them *way*
                            higher than that to be very safe) yet that bug caused failures much more
                            often, and introduced a method of failure that Rabin-Miller *can't*
                            produce, (Rabin-Miller can't ever declare a prime to be composite, but
                            it can, very rarely, declare a composite to be probably prime,) which is
                            why this failure mode was particularly unexpected to a lot of people,
                            and caused numbers above a certain size to fail mysteriously and
                            sporadically.

                            Note: When I was informed of this bug in my implementation of my
                            JacobiSymbol[ x] function, (it didn't affect my primality testing, which
                            has always worked properly,) I was ashamed that I wasn't able to release
                            a fix until I got home from work later that day. Sun, on the other
                            hand, took *3 years* to release a fix. (This bug was present in Java
                            versions 1.3 through 1.4.2.)

                            [4] Of course, some of their "failures" were due to intentional
                            attempts of people to corrupt their results, so exact probabilities of
                            failure are impossible to come by. There were interesting stories,
                            though, of *all* or *almost all* of the work units returned by several
                            well-meaning participants returning incorrect results. The SETI team
                            contacted these people and were able to verify that their computers'
                            floating-point units were indeed failing. (Sometimes subtly enough to
                            not make everything crash, but enough to make all extended calculations
                            wrong.) This is interesting and probably indicates that the frequency
                            of subtly or explicitly broken processors in the world is far less than
                            10^-9, setting a bound for what reliability we might expect for
                            primality testing.

                            [5] Don't laugh. This happened on this very list. A few years ago,
                            an intrepid researcher stated that they had been running a computer for
                            3+ years to find the factors of one of the RSA factoring challenge
                            numbers. Awesome persistence! And then one day it beeped! (I can't
                            imagine the excitement!) He announced to this list that he had
                            submitted his solution to RSA and was awaiting confirmation of the
                            factors, and he wasn't sure if the numbers were factors. I asked him
                            the obvious question, "did you multiply the two factors together and did
                            they come up with the original number?" The next day, with a leaden
                            heart, he responded to me and indicated that he had accidentally cut off
                            a digit when pasting in the original RSA number to his factoring
                            program. It still hurts me to think about it.

                            --
                            Alan Eliasen
                            eliasen@mindspring. com
                            http://futureboy. us/






                            [Non-text portions of this message have been removed]
                          • Alan Eliasen
                            ... I didn t mean to imply in any way that pseudoprime tests were magically free of the same kinds of hardware/software/ human error. Of course they re not,
                            Message 13 of 20 , Apr 25 3:01 PM
                            • 0 Attachment
                              On 04/25/2010 07:14 AM, Jens Kruse Andersen wrote:
                              > Your numbers apparently assume there are no hardware, software,
                              > operator or other errors when the pseudoprime tests are performed.
                              > But they can have the same type of errors as primality proofs.

                              I didn't mean to imply in any way that pseudoprime tests were
                              magically free of the same kinds of hardware/software/ human error. Of
                              course they're not, but perhaps I didn't state this clearly enough. The
                              limiting factor for reliability is always going to be the weakest link
                              in the chain (possibly the person cutting-and-pasting in the number.)

                              > Fast pseudoprime test programs may implement more complicated
                              > algorithms with greater risk of programming error, and may be written
                              > in assembler that tends to produce more errors for human programmers
                              > (partly because of increassed source code size), so it may be better to
                              > run a smaller number of pseudoprime tests with a slow simple program
                              > than a large number with a fast program.

                              A Rabin-Miller algorithm can be written so simply that testing
                              against the trivial, non-optimized version is likely always beneficial.
                              I definitely agree that the probability of undetected programming
                              errors increases with source size and complexity of algorithms, though.
                              We've all seen how hard it is to get even the multiplication of two
                              integers always correct using something like an FFT algorithm. From the
                              notes of the GMP project, many of these errors are even due to broken
                              compilers!

                              > If you run the same type of pseudoprime test with multiple supposedly
                              > independent programs then some things risk causing the same error,
                              > for example:
                              > An error in a text read by multiple programmers.
                              > A systematic hardware error affecting operations likely to be performed
                              > by different programs implementing the same algorithm.
                              > A tricky part of the implementation which may fool multiple programmers.
                              > An error in a software library or other routine used by multiple programs.

                              Don't forget the probability of cutting-and-pasting the same wrong
                              number into all of those programs! Or receiving the wrong number due to
                              transmission errors.

                              The concept of independent verification reducing probability of error
                              rests on the idea that you don't have some sort of systematic error. If
                              you *do* have systematic error, no amount of independent validation will
                              improve your answers, and that needs to be understood.

                              > In brief, if you want to increase the "real" chance that a number is
                              > prime then I think you should bet more on independence of as
                              > much as possible (software, hardware, algorithms, people, source
                              > of the number), than on increasing the number of pseudoprime tests.
                              > And aim for not only two independent of something but as many as
                              > possible or practical.

                              Yes, my point was that reduction of error probability below the most
                              likely mode of failure could only be achieved by truly independent
                              tests. This means you have to look very closely indeed at eliminating
                              potential systematic errors. There are many of these failure modes and
                              some of their probabilities are very high. I cited human error or TCP
                              transmission error or mail/web client/server bugs as being
                              high-probability, systematic sources of failure.

                              The probability of these systematic errors probably also increases if
                              the number isn't of a simple form, e.g. 2^12345-1, but rather an
                              uncompressable number like the 300-digit number cited, which has more
                              probability of corruption in transmission, errors in wrapping, or
                              undetected cut-and-paste error, etc.

                              --
                              Alan Eliasen
                              eliasen@...
                              http://futureboy.us/
                            • Ali Adams
                              Alan you will be happy to see this article on the BBC, UK :) http://news.bbc.co.uk/2/hi/technology/8637845.stm Web security attack makes silicon chips more
                              Message 14 of 20 , Apr 26 3:05 AM
                              • 0 Attachment
                                Alan you will be happy to see this article on the BBC, UK :)
                                http://news.bbc.co.uk/2/hi/technology/8637845.stm
                                Web security attack 'makes silicon chips more reliable'
                                ---------------
                                An attack on a widely used web security system could soon help make silicon chips more powerful and reliable.
                                Many websites use cryptographic systems to scramble key data, such as credit card numbers, when customers pay.
                                Scientists have found that by varying the voltage to key parts of a computer's processor, the ability to keep this data secret is compromised.
                                The researchers also discovered that a method that helps chips beat the attack could also make them more reliable.
                                Secure sites
                                Many modern security systems, such as the ones websites use to encrypt the credit card numbers of their customers, are based around a system known as public key cryptography.
                                This uses two keys, one public and one private, to scramble data. One of the most widely used implementations of this is known as RSA authentication.
                                "If data is locked with a public key, it can only be unlocked with the corresponding private key," said Professor Todd Austin, from the electrical engineering and computer science department at the University of Michigan who helped conduct the research.
                                Within 10 years a chip will have transistor failures every day

                                Professor Valeria Bertacco
                                "It's the kind of algorithm you use when you go to a website and you see the little padlock in the lower right hand corner to indicate a secure connection," he said.
                                The keys take the form of large numbers more than 1,000 digits long. Security is ensured because trying to guess a private number by trying all possible combinations would take longer than the age of the universe, using current computer technology.
                                Professor Austin, working with Andrea Pellegrini and Professor Valeria Bertacco, found a much quicker route to guessing the keys by varying the voltage to a processor.
                                "You need to be able to control the voltage to the power source to the device," said Professor Bertacco. "By putting the voltage just below where it should be means the device makes computational mistakes - it suffers temporary transistor failure."
                                The voltage was varied when a target machine was communicating with another machine via the web and the data flying between the two was encrypted using the public key system.
                                "It makes one mistake every now and again," she said. "But we need just a few mistakes."
                                During their test, the three researchers collected 8800 corrupted signatures in 10 hours and then analysed them using software that could call on 81 separate machines to boost its number crunching power.
                                The end result of the research was an attack method that could extract all the parts of a 1024 bit key in about 100 hours.
                                'Error prone'
                                Initially, said Professor Bertacco, the work will lead to improvements in the way the public key security system works to make it less susceptible to such an attack. Future versions of the system will be "salted" with fake values to confuse any attempt to reconstruct a private key.
                                "It's part of the ongoing process of hardening RSA," said Professor Austin.
                                The implications of the research do not stop at security. It is also helping to produce error correction systems that spot when transistors fail and ensure that data is not corrupted as a result.
                                Professor Bertacco said the research would be useful when chips are made of even smaller components than those in use today. The widely-known Moore's Law predicts that the number of transistors on a given size of silicon wafer doubles roughly every 18 months.
                                Often that doubling is due to the transistors on the chip getting smaller. The transistors on Intel's most up to date desktop computers are about 32 nanometres in size.
                                Intel has said that it expects to soon start producing chips with components 22 and 16nm wide. A nanometre is a billionth of a metre.
                                However, as components get smaller they can get less reliable and need error checking and correction software to help cope with any errors that get introduced.
                                "Our mainstream research in this area is to make microchips operate correctly even in the face of transistor failure," she said. "Within 10 years a chip will have transistor failures every day. As transistors get smaller so they are more prone to failure."
                                ---------------

                                 
                                Ali
                                God > infinity




                                ________________________________
                                From: Alan Eliasen <eliasen@...>
                                To: Ali Adams <alipoland@...>; Prime Number <primenumbers@yahoogroups.com>
                                Sent: Sun, April 25, 2010 3:59:17 PM
                                Subject: [PrimeNumbers] Strong pseudosermon (was 62-digit IsPrime)

                                 
                                On 04/21/2010 09:01 PM, Ali Adams wrote:
                                > Greetings all,
                                >
                                > Can someone please help with validating if the following 62-digit
                                > number is a prime or not:
                                >
                                > 1110819595668080516 5653650502135350 6057690906175754 64617311659
                                >
                                > I am aware of IsProbablePrime but need a definite primality test.

                                (in another posting, asked)
                                > What about this 309-digit number?

                                >259336006801222696 0141827989906545 7702032918541745 1253947966996203 3747780569585929 9412847062270835 2120230964321433 3705453431960894 5822253023887835 9827583627468563 4622332103985890 9085250794700726 5127498998595582 3067653695374111 7527587085881465 1979141558307396 3161542913121427 0318567530145291 6463755740936626 397

                                (Which is not even probable-prime, as one single strong pseudoprime
                                test shows.)

                                I'm about to sermonize, so be prepared, my brothers and sisters.

                                Whenever someone says they "need" a definite primality test, they, by
                                their actions, usually prove that they don't understand that a
                                primality-proving test, by itself, is not an infallible proof of
                                primality (in the real world) and they rarely if ever do the correct
                                thing to decrease the probability of error, giving them a result that's
                                effectively no more certain than a working pseudoprime test (and, by
                                probabilities and failure modes that I'll cite here, probably quite a
                                bit weaker. And by "quite a bit" I mean 20 orders of magnitude, easy.)

                                Note: Keep in mind that all of this analysis applies in the *real
                                world*, not to some perfect, fictitious mathematical abstraction where
                                programmers don't make errors and hardware never fails and you can
                                ignore most algorithms and divide out big pesky constants because the
                                ideal theoretical asymptotic performance is always what you get when you
                                run programs.

                                It is simple to implement and perform a strong pseudoprime test in
                                which the probability that a randomly-chosen composite number is
                                mistakenly stated to be prime is so low that it would never happen in
                                your lifetime.[1] Not only that, you can trivially make the probability
                                of this mistake so low that you could test trillions of numbers every
                                second for the lifetime of the universe, and the probability of *any* of
                                those tests failing are still astronomically low, and that's even taking
                                the old ultra-conservative bound that as many as 1/4 of strong
                                pseudoprime tests can fail. See citations in the link below for numbers
                                about how conservative that estimate really is.

                                Many people have pointed out over the years that the probability that
                                your *hardware* or *software* fails (say, due to a high-energy particle
                                passing through your processor, or random thermal drift of electrons[2] ,
                                or a bad transistor) during your primality test or primality "proof"
                                becomes rapidly much, much higher than the probability that, say, a
                                Rabin-Miller test incorrectly declares a composite to be prime.
                                Depending on the size of the number, this is true *even if you only do
                                one pass of a Rabin-Miller test*, and the probability of error in the
                                algorithm is far less than the probability of hardware failure, possibly
                                by hundreds or thousands of tens of thousands of *orders of magnitude*!

                                It may also be far, far more probable that the number you meant to
                                test or the reply (with certificate) was corrupted during communication
                                with someone else. For example, TCP only has a 16-bit checksum, and can
                                miss, say, two single-bit errors separated by 16 bits, for a possible
                                undetected error rate of 2^-16, or 1/65536 in a noisy channel. (Other
                                error-correcting algorithms apply to communication across the internet,
                                and most channels have fairly good s/n ratios so the end-to-end
                                probability of error is luckily usually lower than this.)

                                There's also the probability that Yahoo does something stupid with
                                the long line and cuts off a digit or something. (Some would say the
                                probability of Yahoo's software doing something stupid when adding their
                                cruft to a posting is 1, but that's being mean. Their software isn't
                                really that bad. The probability is actually 1-epsilon.)

                                To see approximate probabilities of the Rabin-Miller test failing
                                for certain number sizes, even with a *single* round of tests, see:

                                http://primes. utm.edu/notes/ prp_prob. html

                                Those probabilities of failure, especially for large numbers, are
                                mind-bogglingly small! Far smaller than the probability of some other
                                source of error.

                                Note that for the 309-digit number posted above, the probability of a
                                strong pseudoprime test mistakenly returning "prime" for a composite
                                number is less than 5.8*10^-29. Is that probability higher than other
                                probabilities we've already listed? (And note that since a strong
                                pseudoprime test easily catches that the number is actually composite.
                                No matter what base you choose. I'm sure that many here wouldn't
                                hesistate to give a cash reward for anyone who can find *any* prime base
                                smaller than the number for which a strong pseudoprime test fails. I
                                will initially offer all the money in my wallet. And I haven't even
                                tried to look for one. I'm that confident in the probabilities, or my
                                wallet is at its usual sad level.)

                                It's hard to approximate the probability that a particular piece of
                                software or hardware or communication will fail, but you can never
                                expect your primality "proof" to be any stronger than the most likely
                                error in the chain.

                                If you really *need* a prime number, you *must* then be sure to take
                                the certificate produced by one program's prime number proof *and verify
                                this certificate* , presumably with different software on a different
                                machine and hardware architecture! (I feel I should repeat this
                                sentence many times!) And then verify it again with another piece of
                                software! If you *don't* do this, the probability that the primality
                                "proof" is in error is approximately the probability of the thing most
                                likely to go wrong in the entire chain. (Again, maybe cosmic rays,
                                software bugs, Pentiums, race conditions in software, power glitches,
                                probability of human error, etc.) The primality certificate gives you a
                                way of verifying (without performing the whole proof again) that the
                                proof is indeed valid for that number.

                                However, there's nothing that prevents even the *verification* of the
                                certificate from having a similar unlikely failure! (Or similar
                                implementation bugs.)[3] Assuming the implementations and hardware are
                                completely independent, the best you can do is to multiply the
                                probability of failure in both systems. Thus, if the probability of
                                failure in each system independently is 10^-32, then the probability
                                that *both* fail completely independently but in a way that gives the
                                same flawed result is at best 10^-64. Look at the URL cited above and
                                compare that to the probability of a probable-prime test failing. Is it
                                larger or smaller? If the probability is larger, the likelihood of your
                                "proof" being better than a pseudoprime test is purely illusory.

                                I'm not sure what the exact failure probability for a single
                                instruction is in modern hardware, but it's almost certainly *not*
                                better than 10^-32. (Multiply this by the number of instructions
                                required to perform the calculation. )

                                One of the best sources of information I've seen about actual failure
                                rates in installed hardware was done by the SETI@Home team, (which had
                                the world's largest distributed supercomputer at some points) which
                                cited detected error rates in returned work units (each work unit was
                                given to at least two people for validation, so errors could be
                                detected.) I don't remember the exact numbers, but the actual failure
                                rate appeared to be many, many orders of magnitude lower than 10^-32. I
                                don't seem to be able to find these stats at the moment. Does anyone
                                have the link?[4]

                                If you haven't verified a primality certificate independently to
                                decrease your probability of hardware or software error, (and there is
                                *always* a non-zero probability of error on real physical hardware and
                                where humans are involved, even in a primality "proof") you're likely
                                not doing any better than a probabilistic prime test, in which the
                                unreliability of hardware, software, communications, and humans, rapidly
                                become the limiting factors.

                                You're also much more likely that some random mischiefmaker will say,
                                "sure it's prime" when it's not because they're annoyed with you for not
                                doing these simple tests yourself, and for refusing to take advice on
                                tools that will do the work for you. But if people didn't refuse to
                                listen, I wouldn't get a chance to gratuitously sermonize.

                                If you didn't verify the primality certificate as many ways as
                                possible, you clearly can't claim you understand that you "need" a prime
                                number, and don't really understand the probability of all the different
                                ways that failure could have occurred. Do you even know that the
                                primality certificate that was posted here was valid for the number you
                                gave, or did someone maybe miss the last digit digit during
                                cut-and-paste? Do you even know that the certificate was anything but a
                                cat walking across a keyboard? The probability that *you*
                                cut-and-pasted the number incorrectly is orders of magnitude higher than
                                probability of failure of a pseudoprime algorithm.[5]

                                Homework: 1.) Can anyone else estimate probabilities of certain
                                types of errors? What do you think are the probabilities of various
                                failure modes in posting "prove this number prime for me" to a public
                                group? Do any of those probabilities exceed that of failure of a
                                probabilistic test? (Hint: The answer is yes.)

                                2.) Even if you get a provable prime number handed down impeachably
                                from the ghost of Fermat, what's the probability that when you're
                                *using* this prime number that something goes wrong in those
                                calculations? (Hint: Weakest link in the chain.)

                                ============ ====

                                Footnotes:

                                [1] I intentionally state "randomly-chosen" because there are ways
                                to generate a very sparse set of numbers that can fool one of these
                                tests if you know in advance the bases it's going to test, which is why
                                most Rabin-Miller tests choose some bases randomly if there is the
                                potential for adversarial assault. See, for example:
                                François Arnault, Constructing Carmichael numbers which are strong
                                pseudoprimes to several bases, Journal of Symbolic Computation, v.20
                                n.2, p.151-161, Aug. 1995

                                [2] There's a certain nonzero probability that an electron will go
                                "the wrong way" and potentially tunnel "backwards" through a potential
                                barrier. As voltages used in processors get lower, and as gates get
                                smaller and the number of electrons required to switch a gate get fewer,
                                this becomes increasingly more probable. (I don't have my "Feynman
                                Lectures on Computation" at hand or I'd post some equations for this
                                probability. Very highly recommended!
                                http://tinyurl. com/9q4p8o )

                                [3] Multiple programmers creating similar software bugs is more
                                common than you might think. Sun's Java implementation had a famous bug
                                in their calculation of the Jacobi symbol, which caused
                                primality-testing routines to fail:

                                http://bugs. sun.com/bugdatab ase/view_ bug.do?bug_ id=4624738

                                When I implemented a function to calculate the Jacobi symbol in my
                                Frink programming language ( http://futureboy. us/frinkdocs/ ) I
                                initially created a similar bug, because my algorithm didn't work for
                                some negative numbers either. I didn't use Sun's algorithm, but I can
                                see how they went wrong. The algorithm often cited on the web, in
                                several number theory books, etc. needs preconditioning to handle
                                negative numbers correctly. Even more insanely, when evaluating
                                negative numbers, Java uses different sign conventions for the %
                                (modulus) operator (used for "int" values) and the BigInteger.mod( n)
                                function! It's easy to see how Sun even confuses their own programmers.

                                This bug in Sun's implementation caused some significant pain. Their
                                primality testing was originally a probabilistic Rabin-Miller test (with
                                probabilities that can easily be set that it won't return a wrong result
                                during the lifetime of the universe, and many users set them *way*
                                higher than that to be very safe) yet that bug caused failures much more
                                often, and introduced a method of failure that Rabin-Miller *can't*
                                produce, (Rabin-Miller can't ever declare a prime to be composite, but
                                it can, very rarely, declare a composite to be probably prime,) which is
                                why this failure mode was particularly unexpected to a lot of people,
                                and caused numbers above a certain size to fail mysteriously and
                                sporadically.

                                Note: When I was informed of this bug in my implementation of my
                                JacobiSymbol[ x] function, (it didn't affect my primality testing, which
                                has always worked properly,) I was ashamed that I wasn't able to release
                                a fix until I got home from work later that day. Sun, on the other
                                hand, took *3 years* to release a fix. (This bug was present in Java
                                versions 1.3 through 1.4.2.)

                                [4] Of course, some of their "failures" were due to intentional
                                attempts of people to corrupt their results, so exact probabilities of
                                failure are impossible to come by. There were interesting stories,
                                though, of *all* or *almost all* of the work units returned by several
                                well-meaning participants returning incorrect results. The SETI team
                                contacted these people and were able to verify that their computers'
                                floating-point units were indeed failing. (Sometimes subtly enough to
                                not make everything crash, but enough to make all extended calculations
                                wrong.) This is interesting and probably indicates that the frequency
                                of subtly or explicitly broken processors in the world is far less than
                                10^-9, setting a bound for what reliability we might expect for
                                primality testing.

                                [5] Don't laugh. This happened on this very list. A few years ago,
                                an intrepid researcher stated that they had been running a computer for
                                3+ years to find the factors of one of the RSA factoring challenge
                                numbers. Awesome persistence! And then one day it beeped! (I can't
                                imagine the excitement!) He announced to this list that he had
                                submitted his solution to RSA and was awaiting confirmation of the
                                factors, and he wasn't sure if the numbers were factors. I asked him
                                the obvious question, "did you multiply the two factors together and did
                                they come up with the original number?" The next day, with a leaden
                                heart, he responded to me and indicated that he had accidentally cut off
                                a digit when pasting in the original RSA number to his factoring
                                program. It still hurts me to think about it.

                                --
                                Alan Eliasen
                                eliasen@mindspring. com
                                http://futureboy. us/






                                [Non-text portions of this message have been removed]
                              • Jack
                                Alan, Thank you for your thoughtful and considerate reply to the original poster, and to the group in general. I often have questions that I d like to post
                                Message 15 of 20 , Apr 26 6:30 AM
                                • 0 Attachment
                                  Alan,

                                  Thank you for your thoughtful and considerate reply to the original poster, and to the group in general. I often have questions that I'd like to post here but do not because I fear being publicly flamed for my ignorance.


                                  --- In primenumbers@yahoogroups.com, Alan Eliasen <eliasen@...> wrote:
                                  >
                                  > On 04/21/2010 09:01 PM, Ali Adams wrote:
                                  > > Greetings all,
                                  > >
                                  > > Can someone please help with validating if the following 62-digit
                                  > > number is a prime or not:
                                  > >
                                  > > 11108195956680805165653650502135350605769090617575464617311659
                                  > >
                                  > > I am aware of IsProbablePrime but need a definite primality test.
                                  >
                                  > (in another posting, asked)
                                  > > What about this 309-digit number?
                                  >
                                  >


                                  <snip>
                                • djbroadhurst
                                  ... Jens, as per usual, put his finger on the core of this debate. In primality proving, we subject ourselves to two disciplines: 1) do not proclaim a proof if
                                  Message 16 of 20 , Apr 30 7:13 PM
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                                    --- In primenumbers@yahoogroups.com,
                                    "Jens Kruse Andersen" <jens.k.a@...> wrote:

                                    > It will often be more important to an audience of an announced prime
                                    > that you say "Trusted program X proved primality" than you argue
                                    > about the microscopic risk that something went wrong in all the
                                    > pseudoprime tests.

                                    Jens, as per usual, put his finger on the core of this debate.

                                    In primality proving, we subject ourselves to two disciplines:

                                    1) do not proclaim a proof if you cannot understand the proof method

                                    2) take reasonable precautions that your claim has not been
                                    vitiated by egregious soft/hard-ware errors.

                                    Alan's points are well put. Yet he has missed the essential
                                    gravamen of Jens' dictum

                                    > in practice the negative consequences of an alleged prime
                                    > being composite are usually so small

                                    No-one suffers if a cosmic ray hits your computer during a test.
                                    No-one suffers if George's FFT's screw up during that test.
                                    We are trying to be as careful and honest as humanly possible.
                                    The pursuit of excellence is a greater cause than its achievement.

                                    David
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