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

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  • Jud McCranie
    ... I don t think so. It will stop in a finite time. [Non-text portions of this message have been removed]
    Message 1 of 21 , Nov 2, 2003
      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.
      >One definition "operationally" of "Infinity" is the non-stopping Turing
      >machine.

      I don't think so. It will stop in a finite time.


      [Non-text portions of this message have been removed]
    • John Dilick
      Just a quick comment, then back I go... ... You do realize that A000945 is NOT the sequence we have been discussing (namely, the primorial plus one series),
      Message 2 of 21 , Nov 3, 2003
        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
      • John Dilick
        I hate when my brain gets ahead of my fingers... ... in ... A(x)=THE SMALLEST PRIME FACTOR of the product of all of the previous terms. John
        Message 3 of 21 , Nov 3, 2003
          I hate when my brain gets ahead of my fingers...

          > proof. In A000945, A(1)=2, A(x)=the product of all of the previous terms
          in
          > the series plus 1.

          A(x)=THE SMALLEST PRIME FACTOR of the product of all of the previous terms.

          John
        • 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 4 of 21 , Nov 3, 2003
            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 5 of 21 , Nov 3, 2003
              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 6 of 21 , Nov 3, 2003
                > 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 7 of 21 , Nov 3, 2003
                  >>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 8 of 21 , Nov 3, 2003
                    --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 9 of 21 , Nov 3, 2003
                      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 10 of 21 , Nov 3, 2003
                        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 11 of 21 , Nov 4, 2003
                          > 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 12 of 21 , Nov 4, 2003
                            > 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 13 of 21 , Nov 4, 2003
                              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 14 of 21 , Nov 4, 2003
                                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 15 of 21 , Nov 4, 2003
                                  Mr Bagula has been accused of writing "incomprehensible gibberish" before.

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

                                  Don't feed the trolls.


                                  __________________________________________________
                                  Virus checked by MessageLabs Virus Control Centre.
                                • 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 16 of 21 , Nov 4, 2003
                                    --- 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 17 of 21 , Nov 4, 2003
                                      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 18 of 21 , Nov 4, 2003
                                        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 19 of 21 , Nov 4, 2003
                                          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 20 of 21 , Nov 5, 2003
                                            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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