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Re: [PrimeNumbers] Infinite primes-> a Turing Machine prime sieve that never stops?

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  • Roger Bagula
    The sequences related to Euclid s proof are: A000945 A000946 A002585 A005265 A005266 A051342 ( there are several new ones as well) A00945 is just the first
    Message 1 of 21 , Nov 3, 2003
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      The sequences related to Euclid's proof are:
      A000945 A000946 A002585 A005265 A005266 A051342
      ( there are several new ones as well)

      A00945 is just the first "related" sequence.

      But there have to be two schools of thought
      since "proof" isn't working.
      1) Euclid's school infinite primes exist
      2) modern school of thought : infinite primes don't exist

      Unless you can "conclusively" prove no infinite prime exists...
      I've seen nothing like that in the posts.
      It can be "argued" both ways. Paul Leyland
      is good at arguing, but lacks conclusive proofs to go
      with it.

      It is better to accept that there is more than one possible analytic for
      primes
      than make "war" over it.
      As I said wisdom is a better option.
      John Dilick wrote:

      >Just a quick comment, then back I go...
      >
      >
      >
      >>In working with the Euclid proof primes
      >>I found that any reasonable number past about 30 primes in the
      >>product almost never stops in the factoring step.
      >>43 primes in the product was the limit given in the A000945 definition as
      >>being a computing limit.
      >>
      >>
      >
      >You do realize that A000945 is NOT the sequence we have been discussing
      >(namely, the primorial plus one series), right? A000945 is a recursive
      >sequence, that defines itself, using a construction similar to the Euclid
      >proof. In A000945, A(1)=2, A(x)=the product of all of the previous terms in
      >the series plus 1.
      >
      >The primorials (that is, the product of all primes less than or equal to x)
      >plus one is the series generated by the classic Euclid proof, and those HAVE
      >been factored completely well beyond the 43rd prime. Also, there is a
      >gigantic difference between "can't compute this number with the current
      >programs and resources" and "can't compute this number under any
      >circumstances". Wagstaff stopped the computation at the 43rd prime in
      >*1993*. 10 years has made a huge difference in both hardware and software,
      >enough so that even I have factored the primorial+1 series well beyond that
      >point. I'm sure others with a better grasp on both theory and programming
      >have taken it well beyond the point I have, as well.
      >
      >
      >
      >>It is reasonable to assume that an infinite prime might exist ,
      >>but could never be computed.
      >>
      >>
      >
      >Roger, did you not read Paul Leyland's reply to you, the one that started
      >with "What is Infinity?". He explained far better than I have why the
      >notion of an infinite prime is absurd.
      >
      >If you're not going to bother reading the replies, then why are you posting?
      >
      >John
      >
      >
      >
      >
      >Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
      >The Prime Pages : http://www.primepages.org/
      >
      >
      >
      >Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
      >
      >
      >
      >
      >

      --
      Respectfully, Roger L. Bagula
      tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
      URL : http://home.earthlink.net/~tftn
      URL : http://victorian.fortunecity.com/carmelita/435/





      [Non-text portions of this message have been removed]
    • Jud McCranie
      ... A prime that is infinitely large? Show me. [Non-text portions of this message have been removed]
      Message 2 of 21 , Nov 3, 2003
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        At 12:43 PM 11/3/2003, Roger Bagula wrote:
        >Dear Jud McCranie,
        >Don't be tiresome.
        >It is a "standard" definition.
        >Look it up.
        >
        > > At 12:51 PM 11/2/2003 -0800, Roger Bagula wrote:
        > > >It is reasonable to assume that an infinite prime might exist ,

        A prime that is infinitely large? Show me.


        [Non-text portions of this message have been removed]
      • Andy Swallow
        ... I m sorry Roger, but proof *is* working. It s just your grasp of the concepts that s causing the bother. I m truly amazed that you still can t accept the
        Message 3 of 21 , Nov 3, 2003
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          > But there have to be two schools of thought
          > since "proof" isn't working.
          > 1) Euclid's school infinite primes exist
          > 2) modern school of thought : infinite primes don't exist

          I'm sorry Roger, but proof *is* working. It's just your grasp of the
          concepts that's causing the bother. I'm truly amazed that you still
          can't accept the Euclid proof. You seem to have a few people arguing
          against you, so can you please start producing coherent arguments about
          what the problem is meant to be?

          Andy
        • Jud McCranie
          ... It is not reasonable to assume that. Are you assuming that because there are an infinite number of prime numbers that one of them is infinite? That is
          Message 4 of 21 , Nov 3, 2003
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            >>At 12:51 PM 11/2/2003 -0800, Roger Bagula wrote:
            >> >It is reasonable to assume that an infinite prime might exist ,
            >>>but could never be computed.

            It is not reasonable to assume that. Are you assuming that because there
            are an infinite number of prime numbers that one of them is infinite? That
            is false. There are an infinite number of primes, but each of them is finite.

            Also, there is a proof "Every Prime has a succinct certificate", by Vaughn
            Pratt IIRC, which says that every prime can be proven so in a finite number
            of steps/time.
          • Nathan Russell
            ... Equally, there are infinitely many integers, or even numbers, but there is no one infinite integer - it is simply the case that for any integer n, you can
            Message 5 of 21 , Nov 3, 2003
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              --On Monday, November 03, 2003 10:32 PM -0500 Jud McCranie <j.mccranie@...> wrote:

              >
              >>> At 12:51 PM 11/2/2003 -0800, Roger Bagula wrote:
              >>> > It is reasonable to assume that an infinite prime might exist ,
              >>>> but could never be computed.
              >
              > It is not reasonable to assume that. Are you assuming that because there
              > are an infinite number of prime numbers that one of them is infinite? That
              > is false. There are an infinite number of primes, but each of them is finite.

              Equally, there are infinitely many integers, or even numbers, but there is no one infinite integer - it is simply the case that for any integer n, you can find a successor n+1 which is also an integer.

              Every prime also has a successor - there is no largest prime (whereas there IS a largest member of any finite set, for example there is a tallest person in the world, or a smallest planet in the solar system). However, since primes are defined as being a subset of the positive integers, there is no infinite prime.

              Does that make sense, Roger?

              Regards,
              Nathan
            • Jud McCranie
              ... Yes, and there is another thing which sometimes causes confusion, which might be at work here. That is the difference between something being unbounded
              Message 6 of 21 , Nov 3, 2003
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                At 10:37 PM 11/3/2003, Nathan Russell wrote:

                >Equally, there are infinitely many integers, or even numbers, but there is
                >no one infinite integer

                Yes, and there is another thing which sometimes causes confusion, which
                might be at work here. That is the difference between something being
                "unbounded" and it being "infinite". The size of integers/primes is
                unbounded but that doesn't mean that the size can be infinite. Every
                particular integer/prime is finite. There are an infinite number of them.
              • Jud McCranie
                ... Any particular prime - no matter how large - can be computed in a finite amount of time. All primes below any finite limit, no matter how large, can be
                Message 7 of 21 , Nov 3, 2003
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                  At 12:43 PM 11/3/2003, Roger Bagula wrote:

                  >> >It is reasonable to assume that an infinite prime might exist ,
                  >>>but could never be computed.
                  >>>One definition "operationally" of "Infinity" is the non-stopping Turing
                  >>>machine.

                  Any particular prime - no matter how large - can be computed in a finite
                  amount of time. All primes below any finite limit, no matter how large,
                  can be computed in a finite time. But you can't explicitly compute all
                  primes in a finite time.

                  I've been talking too much, but I'm trying to help clear up the
                  misunderstanding.
                • Paul Leyland
                  ... Very well, time for another argument. It would be helpful if you respond (concisely and politely please) to each of the questions below. First my thesis:
                  Message 8 of 21 , Nov 4, 2003
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                    > From: Roger Bagula [mailto:tftn@...]

                    > The sequences related to Euclid's proof are:
                    > A000945 A000946 A002585 A005265 A005266 A051342
                    > ( there are several new ones as well)
                    >
                    > A00945 is just the first "related" sequence.
                    >
                    > But there have to be two schools of thought
                    > since "proof" isn't working.
                    > 1) Euclid's school infinite primes exist
                    > 2) modern school of thought : infinite primes don't exist
                    >
                    > Unless you can "conclusively" prove no infinite prime exists...
                    > I've seen nothing like that in the posts.
                    > It can be "argued" both ways. Paul Leyland
                    > is good at arguing, but lacks conclusive proofs to go
                    > with it.

                    Very well, time for another argument. It would be helpful if
                    you respond (concisely and politely please) to each of the
                    questions below.

                    First my thesis: infinite primes do not exist. The set of
                    primes is infinite (because it has cardinality aleph_0).

                    Consider carefully the difference between these two statements.
                    The first says that, because infinity is not an integer and primes
                    are integers, the concept "infinite prime" is meaningless. The
                    second says that the set of primes can be put in one-to-one
                    correspondence with the set of integers >0.

                    Yuo do not seem to like the classical proof, so let's prove the
                    number of primes (i.e the cardinality of the set of primes) is
                    infinite by another approach.

                    Do you agree that the set of natural numbers is infinite?

                    That is, do you believe that given a particular integer N, we can
                    always find a larger integer M, no matter how large N may be?

                    Do you agree that, given N, the value M=N+1 is larger than N?

                    Again, if not the game is over because we are not talking about
                    mathematics as understood by almost all the world's mathematicians.

                    Do you agree that each and every integer N > 1 has a unique
                    factorization into primes?

                    I'm concerned only with which primes appear in the factorization,
                    not with the order in which they are given. Numerical order is
                    conventional but it really doesn't matter which order they are in
                    for the arguments below.

                    Next, I am going to define F_N, where N is an integer > 0, to be
                    the value 2^(2^N)+1, so F_1 has the value 5, F_2 has the value 17,
                    F_3 has the value 257, F_4 has the value 65537, and so on.

                    Do you agree that the set of all values F_N is infinite?

                    I claim it is because the N can take any integer value and each
                    value of N yields a value for F_N.

                    Do you agree that all the values for F_N are distinct?

                    I claim they are because 2^N is larger than N, and 2^(2^N) is
                    larger than 2^N. I therefore claim that each F_i is greater
                    than F_j whenever i>j.

                    Do you agree that (x+y)*(x-y) = x^2 - y^2?

                    Now, let's take a look at F_N - 2. By definition of F_N, this
                    quantity is equal to 2^(2^N) - 1. Note that 1 is a square (it is
                    1 squared and that 2^(2^N) is a square (it is 2^(2^(N-1)) squared.)

                    So, do you agree that the following equation is correct?

                    F_N - 2 = (2^(2^(N-1)) + 1) * ( 2^(2^(N-1)) - 1)

                    If so, you will readily conclude that the first term is just the
                    definition of F_(N-1), and the second is just F_(N-1) - 2).

                    Now what does this tell us? (A rhetorical question, no need to
                    answer). It says that F_N - 2 is a multiple of F_(N-1). Of course,
                    we have to be careful. F_i is not defined if i is less than 1, so
                    we have to insist that N >= 2 in the above equation. So lets start
                    with N=2 and work our way up.

                    When N=2, F_2 - 2 = F_1 * (F_1 - 2).

                    This is easy to check:
                    F_2 is 17, F_1 is 5 and, indeed F_2-2 = 15 = 5*3 = F_1 * (F_1 -2).

                    When N=3, F_3 - 2 = 255
                    = F_2 * (F_2 - 2)
                    = 17 * F_1 * (F_1 - 2)
                    = 17 * 5 * 3.

                    When N = 4, F_4 - 2 = 65535
                    = F_3 * (F_3 - 2)
                    = 257 * F_2 * (F_2 - 2)
                    = 257 * 17 * F_1 * (F_1 - 2)
                    = 257 * 17 * 5 * 3.

                    With me so far? Do you agree that we can continue this process
                    as far as we wish, because we can increase the value of N as far as
                    we wish?

                    If not, why not? I can try to make it clearer. If you do agree,
                    let's proceed.

                    I now claim that F_M has no prime factors in common with *any* F_N
                    for which N < M. The reasoning is that F_M - 2 is a multiple of
                    all F_N for which N < M. Thus, any prime factor of F_N yields a
                    remainder of 2 when divided into F_M. All the F_N numbers are odd,
                    so we can discount the prime number 2 as being a factor of any of them.

                    Do you agree with this claim? If not, please explain what you think
                    is wrong with it.

                    Assuming you do agree, you will also agree that I have produced an
                    infinite set of numbers, none of which share a common prime factor.

                    The set of all F_N consists only of integers, so each and every one
                    has at least one prime factor, agreed?

                    But as the set of all F_N has cardinality aleph_0, the same as the
                    cardinality of the set of all integers >=1, AND every member of the
                    set of all F_N is an integer, AND every member of the set of all F_N
                    has at least one prime factor, AND no two members share any prime
                    factors, THEN the cardinality of the set of all prime factors of all
                    the members of the set of all F_N is itself aleph_0.

                    What does the above lengthy and pedantically stated sentence say
                    (another rhetorical question)? It says that we have constructed
                    a set of prime numbers which has cardinality aleph_0. That is,
                    we've found an infinite set of prime numbers. This, I claim, proves
                    my thesis given at the start of this article. Note I do *not* claim
                    that the set of primes I constructed contains all the primes. In fact,
                    we know for certain that it does not, and you should be able to prove
                    that, for instance, neither 2 nor 13 are members of the set.
                    Nonetheless, it is an infinite set.

                    Conclusion: the set of all primes is infinite (has cardinality
                    aleph_0) because at least one set which contains only primes itself
                    has cardinality aleph_0.


                    Paul
                  • Andy Swallow
                    ... Make up your mind Roger, what are you arguing? Are you (a) talking about whether the number of primes is infinite or not, or (b) whether there are some
                    Message 9 of 21 , Nov 4, 2003
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                      > But there have to be two schools of thought
                      > since "proof" isn't working.
                      > 1) Euclid's school infinite primes exist
                      > 2) modern school of thought : infinite primes don't exist
                      >
                      > Unless you can "conclusively" prove no infinite prime exists...
                      > I've seen nothing like that in the posts.

                      Make up your mind Roger, what are you arguing? Are you (a) talking about
                      whether the number of primes is infinite or not, or (b) whether there are
                      some magical new numbers which you decide to call "infinite primes"?
                      You're quite correct of course, nothing in the posts has proved that no
                      infinite prime exists. In the case of (a), your attempts at proof have
                      been less than wonderful, and in the case of (b) none of us have any
                      idea what you mean by infinite prime.

                      I presume you're talking about whether there are infinitely many primes
                      or not, but you still seem to think that this would imply the existence
                      of "infinitely large primes". Not true. There would be *arbitrarily*
                      large primes, but they would always be finite. It's important to
                      understand this. Don't you have any number theory books you can look at
                      for this stuff? Try the first chapter of an introductory book...

                      As a little amusement, suppose that there were only finitely many
                      primes. Then the Euler product form of the zeta function would define an
                      entire function, thus making zeta(s) an entirely different animal.
                      Congratulations Roger, you would have answered the Riemann hypothesis, one
                      way or another.

                      Andy

                      PS "Modern school of thought"? You mean "Roger's school of thought"?
                      Small school then?
                    • Roger Bagula
                      I ve just been trying to get the stuff on primes that calculation allows and develop some tricks like: I found a neat identity last night:
                      Message 10 of 21 , Nov 4, 2003
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                        I've just been trying to get the stuff on primes that calculation allows
                        and develop some "tricks" like:
                        I found a neat identity last night:
                        n!=Product[Prime[i],{i,1,PrimePi[n]}]*Product[Composite[i],{i,1,n-PrimePi[n]}]
                        It allows one to compute the composites nearly directly in Mathematica:
                        (* Composite Product*)
                        p[n_]=n!/Product[Prime[i],{i,2,PrimePi[n]}]
                        digits=200
                        a0=Table[p[n]/p[n-1],{n,2,digits}]
                        (* Composites by sorting out ones and two*)
                        Delete[Delete[Union[a0],1],1]

                        This function works much better than taking all the primes out of a list
                        of integers.

                        You can also get an higher number of primes by the Euclid proof
                        type integer factoring method.
                        Nathan Russell wrote:

                        >--On Monday, November 03, 2003 10:32 PM -0500 Jud McCranie <j.mccranie@...> wrote:
                        >
                        >
                        >
                        >>>>At 12:51 PM 11/2/2003 -0800, Roger Bagula wrote:
                        >>>>
                        >>>>
                        >>>>>It is reasonable to assume that an infinite prime might exist ,
                        >>>>>but could never be computed.
                        >>>>>
                        >>>>>
                        >>It is not reasonable to assume that. Are you assuming that because there
                        >>are an infinite number of prime numbers that one of them is infinite? That
                        >>is false. There are an infinite number of primes, but each of them is finite.
                        >>
                        >>
                        >
                        >Equally, there are infinitely many integers, or even numbers, but there is no one infinite integer - it is simply the case that for any integer n, you can find a successor n+1 which is also an integer.
                        >
                        >Every prime also has a successor - there is no largest prime (whereas there IS a largest member of any finite set, for example there is a tallest person in the world, or a smallest planet in the solar system). However, since primes are defined as being a subset of the positive integers, there is no infinite prime.
                        >
                        >Does that make sense, Roger?
                        >
                        >Regards,
                        >Nathan
                        >
                        >
                        >Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                        >The Prime Pages : http://www.primepages.org/
                        >
                        >
                        >
                        >Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
                        >
                        >
                        >
                        >
                        >

                        --
                        Respectfully, Roger L. Bagula
                        tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
                        URL : http://home.earthlink.net/~tftn
                        URL : http://victorian.fortunecity.com/carmelita/435/





                        [Non-text portions of this message have been removed]
                      • Roger Bagula
                        The concept of infinity is a Platonic ideal. It can t be proven. It is a limiting axiomatic definition (asymptotic). So arguing if such an such types of
                        Message 11 of 21 , Nov 4, 2003
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                          The concept of "infinity"
                          is a Platonic ideal.
                          It can't be proven.
                          It is a limiting axiomatic definition (asymptotic).
                          So arguing if such an such types of infinity
                          exist , makes sense only if the limiting
                          case is assumed to exist in the first place.
                          Mathematics can be defined as two cases:
                          1) numbers where infinity is defined
                          2) numbers where no infinity is defined or definable as
                          being a non-operational definition
                          ( it crashes Mathematica, ha, ha... )
                          It is a philosophical distinction outside of normal mathematics.
                          Called since Gödel's time "metamathematical" statements/ arguments
                          after it was realized that a philosophical threshold
                          existed in certain kinds of proofs.
                          Statements like :
                          3) Aleph1=2^Aleph0
                          are outside of ordinary mathematics.
                          The algebra of transfinites is really a form of
                          metamathematics invented and outside ordinary mathematics.
                          It has it's own axiom system and rules.
                          The idea of an infinite prime would have to be
                          in this context and not in the context of finite primes at all.
                          A lot of people don't like it pointed out that irrational numbers depend on
                          a statement like 3)
                          being accepted as an axiomatic definition.
                          Like Euclid's product proof,
                          Cantor's proof of transfinites is also widely faulted in
                          modern mathematics.
                          The result is you get philosophical
                          and nearly religious in tone arguments
                          in mathematical forms like these egroups.
                          At first I didn't realize that it was that basic a
                          "postulate" , "axiom"
                          or
                          "definition", but it has become clear
                          that it is.
                          In modern science we are talking about a quantum
                          measure limit near 10^(-33) cm, below which
                          what we know of as material and fields can't exist.
                          Much of current discussion on quantum gravity
                          comes down to "finite lattices" of spatial points.
                          So ideas at the integer level become very important.
                          And "infinities" become important on a philosophical
                          basis:
                          do they exist
                          or don't they?
                          Most of our calculations that make up
                          our ordinary lives don't require that any kind of infinity exists.

                          I don't know if any of this answers you list of questions or not.
                          I really don't like to get in such discussions,
                          since math people seem to ignore any philosophical issues by defining
                          them away.
                          Axioms and definitions are an answer to all their thinking problems?
                          As a physical scientist ( chemist, physical scientist)
                          I'm not bound by those rules in my thinking.
                          So I'm glad to say I can think of things outside of those rules.
                          I used to think that the topological approach was an answer
                          but that approach is bounded by set theory as well
                          and the logical problems of set theory seem to crop up
                          in it's proofs of infinite primes too?

                          The basic/ fundamental question seems to be :
                          Is any system that is based on axioms that resolve into
                          Platonic ideals, actually inductively defensible
                          as a form of knowledge?

                          In other words is science more fundamental
                          philosophically than mathematics at a metamathematical level?
                          Paul Leyland wrote:

                          >>From: Roger Bagula [mailto:tftn@...]
                          >>
                          >>
                          >
                          >
                          >
                          >>The sequences related to Euclid's proof are:
                          >>A000945 A000946 A002585 A005265 A005266 A051342
                          >>( there are several new ones as well)
                          >>
                          >>A00945 is just the first "related" sequence.
                          >>
                          >>But there have to be two schools of thought
                          >>since "proof" isn't working.
                          >>1) Euclid's school infinite primes exist
                          >>2) modern school of thought : infinite primes don't exist
                          >>
                          >>Unless you can "conclusively" prove no infinite prime exists...
                          >>I've seen nothing like that in the posts.
                          >>It can be "argued" both ways. Paul Leyland
                          >>is good at arguing, but lacks conclusive proofs to go
                          >>with it.
                          >>
                          >>
                          >
                          >Very well, time for another argument. It would be helpful if
                          >you respond (concisely and politely please) to each of the
                          >questions below.
                          >
                          >First my thesis: infinite primes do not exist. The set of
                          >primes is infinite (because it has cardinality aleph_0).
                          >
                          >Consider carefully the difference between these two statements.
                          >The first says that, because infinity is not an integer and primes
                          >are integers, the concept "infinite prime" is meaningless. The
                          >second says that the set of primes can be put in one-to-one
                          >correspondence with the set of integers >0.
                          >
                          >Yuo do not seem to like the classical proof, so let's prove the
                          >number of primes (i.e the cardinality of the set of primes) is
                          >infinite by another approach.
                          >
                          >Do you agree that the set of natural numbers is infinite?
                          >
                          >That is, do you believe that given a particular integer N, we can
                          >always find a larger integer M, no matter how large N may be?
                          >
                          >Do you agree that, given N, the value M=N+1 is larger than N?
                          >
                          >Again, if not the game is over because we are not talking about
                          >mathematics as understood by almost all the world's mathematicians.
                          >
                          >Do you agree that each and every integer N > 1 has a unique
                          >factorization into primes?
                          >
                          >I'm concerned only with which primes appear in the factorization,
                          >not with the order in which they are given. Numerical order is
                          >conventional but it really doesn't matter which order they are in
                          >for the arguments below.
                          >
                          >Next, I am going to define F_N, where N is an integer > 0, to be
                          >the value 2^(2^N)+1, so F_1 has the value 5, F_2 has the value 17,
                          >F_3 has the value 257, F_4 has the value 65537, and so on.
                          >
                          >Do you agree that the set of all values F_N is infinite?
                          >
                          >I claim it is because the N can take any integer value and each
                          >value of N yields a value for F_N.
                          >
                          >Do you agree that all the values for F_N are distinct?
                          >
                          >I claim they are because 2^N is larger than N, and 2^(2^N) is
                          >larger than 2^N. I therefore claim that each F_i is greater
                          >than F_j whenever i>j.
                          >
                          >Do you agree that (x+y)*(x-y) = x^2 - y^2?
                          >
                          >Now, let's take a look at F_N - 2. By definition of F_N, this
                          >quantity is equal to 2^(2^N) - 1. Note that 1 is a square (it is
                          >1 squared and that 2^(2^N) is a square (it is 2^(2^(N-1)) squared.)
                          >
                          >So, do you agree that the following equation is correct?
                          >
                          >F_N - 2 = (2^(2^(N-1)) + 1) * ( 2^(2^(N-1)) - 1)
                          >
                          >If so, you will readily conclude that the first term is just the
                          >definition of F_(N-1), and the second is just F_(N-1) - 2).
                          >
                          >Now what does this tell us? (A rhetorical question, no need to
                          >answer). It says that F_N - 2 is a multiple of F_(N-1). Of course,
                          >we have to be careful. F_i is not defined if i is less than 1, so
                          >we have to insist that N >= 2 in the above equation. So lets start
                          >with N=2 and work our way up.
                          >
                          >When N=2, F_2 - 2 = F_1 * (F_1 - 2).
                          >
                          >This is easy to check:
                          >F_2 is 17, F_1 is 5 and, indeed F_2-2 = 15 = 5*3 = F_1 * (F_1 -2).
                          >
                          >When N=3, F_3 - 2 = 255
                          > = F_2 * (F_2 - 2)
                          > = 17 * F_1 * (F_1 - 2)
                          > = 17 * 5 * 3.
                          >
                          >When N = 4, F_4 - 2 = 65535
                          > = F_3 * (F_3 - 2)
                          > = 257 * F_2 * (F_2 - 2)
                          > = 257 * 17 * F_1 * (F_1 - 2)
                          > = 257 * 17 * 5 * 3.
                          >
                          >With me so far? Do you agree that we can continue this process
                          >as far as we wish, because we can increase the value of N as far as
                          >we wish?
                          >
                          >If not, why not? I can try to make it clearer. If you do agree,
                          >let's proceed.
                          >
                          >I now claim that F_M has no prime factors in common with *any* F_N
                          >for which N < M. The reasoning is that F_M - 2 is a multiple of
                          >all F_N for which N < M. Thus, any prime factor of F_N yields a
                          >remainder of 2 when divided into F_M. All the F_N numbers are odd,
                          >so we can discount the prime number 2 as being a factor of any of them.
                          >
                          >Do you agree with this claim? If not, please explain what you think
                          >is wrong with it.
                          >
                          >Assuming you do agree, you will also agree that I have produced an
                          >infinite set of numbers, none of which share a common prime factor.
                          >
                          >The set of all F_N consists only of integers, so each and every one
                          >has at least one prime factor, agreed?
                          >
                          >But as the set of all F_N has cardinality aleph_0, the same as the
                          >cardinality of the set of all integers >=1, AND every member of the
                          >set of all F_N is an integer, AND every member of the set of all F_N
                          >has at least one prime factor, AND no two members share any prime
                          >factors, THEN the cardinality of the set of all prime factors of all
                          >the members of the set of all F_N is itself aleph_0.
                          >
                          >What does the above lengthy and pedantically stated sentence say
                          >(another rhetorical question)? It says that we have constructed
                          >a set of prime numbers which has cardinality aleph_0. That is,
                          >we've found an infinite set of prime numbers. This, I claim, proves
                          >my thesis given at the start of this article. Note I do *not* claim
                          >that the set of primes I constructed contains all the primes. In fact,
                          >we know for certain that it does not, and you should be able to prove
                          >that, for instance, neither 2 nor 13 are members of the set.
                          >Nonetheless, it is an infinite set.
                          >
                          >Conclusion: the set of all primes is infinite (has cardinality
                          >aleph_0) because at least one set which contains only primes itself
                          >has cardinality aleph_0.
                          >
                          >
                          >Paul
                          >
                          >
                          >
                          >Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
                          >The Prime Pages : http://www.primepages.org/
                          >
                          >
                          >
                          >Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
                          >
                          >
                          >
                          >
                          >

                          --
                          Respectfully, Roger L. Bagula
                          tftn@..., 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
                          URL : http://home.earthlink.net/~tftn
                          URL : http://victorian.fortunecity.com/carmelita/435/





                          [Non-text portions of this message have been removed]
                        • Paul Jobling
                          Mr Bagula has been accused of writing incomprehensible gibberish before. http://www.geocities.com/CapeCanaveral/Launchpad/5113/fr33.htm Don t feed the
                          Message 12 of 21 , Nov 4, 2003
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                            Mr Bagula has been accused of writing "incomprehensible gibberish" before.

                            http://www.geocities.com/CapeCanaveral/Launchpad/5113/fr33.htm

                            Don't feed the trolls.


                            __________________________________________________
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                          • pakaran42
                            ... I don t know if the concept of infinity can be proven, but statements about transfinite numbers can be. ... Which is irrelevant to the philosophical
                            Message 13 of 21 , Nov 4, 2003
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                              --- In primenumbers@yahoogroups.com, Roger Bagula <tftn@e...> wrote:
                              > The concept of "infinity"
                              > is a Platonic ideal.
                              > It can't be proven.

                              I don't know if "the concept of infinity" can be proven, but
                              statements about transfinite numbers can be.

                              > It is a limiting axiomatic definition (asymptotic).
                              > So arguing if such an such types of infinity
                              > exist , makes sense only if the limiting
                              > case is assumed to exist in the first place.
                              > Mathematics can be defined as two cases:
                              > 1) numbers where infinity is defined
                              > 2) numbers where no infinity is defined or definable as
                              > being a non-operational definition
                              > ( it crashes Mathematica, ha, ha... )

                              Which is irrelevant to the philosophical argument.

                              The number 3^100 cannot be represented exactly in most versions of C,
                              and pi cannot be represented exactly in any language I know of without
                              taking shortcuts. Does this mean neither exists?

                              > It is a philosophical distinction outside of normal mathematics.
                              > Called since Gödel's time "metamathematical" statements/ arguments
                              > after it was realized that a philosophical threshold
                              > existed in certain kinds of proofs.
                              > Statements like :
                              > 3) Aleph1=2^Aleph0
                              > are outside of ordinary mathematics.

                              No, they're just plain false (and I believe that can be shown from
                              ZFC, though I wouldn't want to try from scratch).

                              Assume Aleph1=2^Aleph0. This implies that there is a mapping from the
                              Aleph1 real numbers onto the base 2 logarithms of the Aleph0 integers.
                              However, there are only countably many such logarithms (since there
                              are only countably many positive integers for them to be logarithms
                              of). This is a contradiction (and I'm sure it can be stated much more
                              formally). I would agree that the integers are a proper subset of the
                              base-2 logs of integers, just as the primes, perfect squares, and
                              numbers evenly divisible by 100 are proper subsets of the integers.
                              However, all those sets are countably infinite.

                              > The algebra of transfinites is really a form of
                              > metamathematics invented and outside ordinary mathematics.
                              > It has it's own axiom system and rules.

                              Not quite the case.

                              > The idea of an infinite prime would have to be
                              > in this context and not in the context of finite primes at all.

                              True.

                              > A lot of people don't like it pointed out that irrational numbers
                              depend on
                              > a statement like 3)
                              > being accepted as an axiomatic definition.
                              > Like Euclid's product proof,
                              > Cantor's proof of transfinites is also widely faulted in
                              > modern mathematics.

                              By whom? And which proof? The diagonal argument? Care to name
                              someone who faults it?

                              > The result is you get philosophical
                              > and nearly religious in tone arguments
                              > in mathematical forms like these egroups.
                              > At first I didn't realize that it was that basic a
                              > "postulate" , "axiom"
                              > or
                              > "definition", but it has become clear
                              > that it is.

                              > I really don't like to get in such discussions,
                              > since math people seem to ignore any philosophical issues by defining
                              > them away.
                              > Axioms and definitions are an answer to all their thinking problems?
                              > As a physical scientist ( chemist, physical scientist)
                              > I'm not bound by those rules in my thinking.

                              So if I asked you to design a plastic that would work in a world where
                              the electromagnetic constant was 1/1000 of its present value, you
                              could, without starting from scratch? That's about the equivalent of
                              what you're asking us to do.

                              Regards,
                              Nathan
                            • Andrew Swallow
                              All that is fair enough, although if I was feeling cynical I d read that post as I don t know what infinite primes are, but it sounds nice . Perhaps before we
                              Message 14 of 21 , Nov 4, 2003
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                                All that is fair enough, although if I was feeling cynical I'd read
                                that post as "I don't know what infinite primes are, but it sounds
                                nice".

                                Perhaps before we bother ourselves with what infinite primes might be,
                                we should instead concentrate our efforts on persuading the apparently
                                unpersuadeable that Euclid's proof is correct. I can understand how
                                arguments about Cantor's ideas can arise, since the whole concept of
                                uncountability is a sticky area to get used to. But Euclid's proof?

                                You seem either unwilling or unable to explain what your problem is
                                with this ridiculously simple method. The only post in which you've
                                tried to explain has (purposefully?) pushed the theory to a point
                                where an inexperienced amateur would worry that something was wrong.
                                Those of us with some experience pretty much instantly pointed out
                                just how rubbish your arguments were. Please explain!

                                Andy
                              • Jud McCranie
                                ... That shows that there are -- an infinite number of primes. It does NOT show that there is an infinite prime . That is what we have been trying to
                                Message 15 of 21 , Nov 4, 2003
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                                  At 11:20 AM 11/4/2003, Roger Bagula wrote:
                                  >The proof that has been in discussion is at:
                                  ><http://www.utm.edu/research/primes/notes/proofs/infinite/euclids.html>http://www.utm.edu/research/primes/notes/proofs/infinite/euclids.html

                                  That shows that there are -- an infinite number of primes. It does NOT
                                  show that there is an "infinite prime". That is what we have been trying
                                  to explain to you, and your recent message says

                                  "Infinitely many primes is distinct from an Infinite Prime."
                                • Jud McCranie
                                  ... Again, those are proofs that there are an infinite number of primes. None of us disagree with that. But you are assuming that an infinite number of
                                  Message 16 of 21 , Nov 4, 2003
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                                    At 11:24 AM 11/4/2003, Roger Bagula wrote:
                                    >Other proofs at:
                                    ><http://www.utm.edu/research/primes/notes/proofs/infinite/index.html>http://www.utm.edu/research/primes/notes/proofs/infinite/index.html
                                    >
                                    > *
                                    > <http://www.utm.edu/research/primes/notes/proofs/infinite/topproof.html>Furstenberg's
                                    > Topological Proof (1955)
                                    >
                                    ><http://www.utm.edu/research/primes/notes/proofs/infinite/topproof.html>http://www.utm.edu/research/primes/notes/proofs/infinite/topproof.html
                                    >
                                    > *
                                    > <http://www.utm.edu/research/primes/notes/proofs/infinite/goldbach.html>Goldbach's
                                    > Proof (1730)
                                    >
                                    ><http://www.utm.edu/research/primes/notes/proofs/infinite/goldbach.html>http://www.utm.edu/research/primes/notes/proofs/infinite/goldbach.html

                                    Again, those are proofs that there are an infinite number of primes. None
                                    of us disagree with that. But you are assuming that an infinite number of
                                    primes implies an "infinite prime", and your own recent message says that
                                    isn't the case


                                    "A semantic distinction needs to be made:

                                    Infinitely many primes is distinct from an Infinite Prime.

                                    Although one seems to imply the other,
                                    they really involve two separate cases. "




                                    [Non-text portions of this message have been removed]
                                  • Paul Leyland
                                    Roger Bagula wrote: [Philosophical meandering deleted.] ... No, it does not. Please post again, including my text and at the appropriate point intersperse
                                    Message 17 of 21 , Nov 5, 2003
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                                      Roger Bagula wrote:

                                      [Philosophical meandering deleted.]

                                      > I don't know if any of this answers you list of questions or not.

                                      No, it does not. Please post again, including my text and at
                                      the appropriate point intersperse your answer to each specific
                                      question that I asked.

                                      > I really don't like to get in such discussions,

                                      That is becoming ever more clear as time goes by.

                                      > since math people seem to ignore any philosophical issues by
                                      > defining them away. Axioms and definitions are an answer
                                      > to all their thinking problems?

                                      Close, but no cigar. Axioms and definitions are the foundations
                                      of mathematical thinking. Rigorously correct logical arguments
                                      are the building blocks placed on the foundations. If you wish to
                                      be considered to be doing mathematics, please use clear and precise
                                      logic.

                                      > As a physical scientist ( chemist, physical scientist)
                                      > I'm not bound by those rules in my thinking.

                                      Ah, so you're not a mathematician and you are not interested in
                                      participating in mathematics. Why, then, are you making so much
                                      noise in an indubitably mathematical forum?

                                      FWIW, my background is in the physical sciences. I have a BA in
                                      chemistry from Oxford and my DPhil was for research in molecular
                                      spectroscopy. I personally don't regard that as an obstacle to
                                      contributing in a small way to a mathematical subject. I'm not
                                      bound by the rules of mathematics any more than you are, but I
                                      choose to follow them when communicating with mathematicians. If
                                      you wish to converse with practitioners of other fields of study,
                                      please do so but, please, do it in a relevant forum elsewhere and
                                      use their rules to do so. Again, FWIW, I'm quite happy to talk
                                      about quantum field theory or geometrodynamics, but not here.

                                      > In other words is science more fundamental
                                      > philosophically than mathematics at a metamathematical level?

                                      A very good question and one well worth discussing, but not here.
                                      It is not (IMO, the moderators may disagree) relevant to the
                                      advertised aims of the forum.


                                      Paul
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