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Re: [PrimeNumbers] Finitude of primes

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  • Chris Caldwell
    ... Even primes... Primes divisible by three... Primes of the form x^2-1 (or x^n-1)...
    Message 1 of 14 , Oct 1, 2003
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      At 01:33 PM 10/1/2003 +0000, mad37wriggle wrote:

      >Are there any types/forms of primes of which there are definitely
      >only a finite number? I'm sure there must be, but can't think of
      >any off the top of my head. I'd be interested to see the proofs of
      >finitude...

      Even primes...
      Primes divisible by three...
      Primes of the form x^2-1 (or x^n-1)...
    • mad37wriggle
      OK let me attempt to rephrase that so as to avoid trivial solutions... Are there any types/forms of primes of which there are definitely only a finite ( 1
      Message 2 of 14 , Oct 1, 2003
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        OK let me attempt to rephrase that so as to avoid "trivial"
        solutions...

        Are there any types/forms of primes of which there are definitely
        only a finite (>1 and taken over all possible primes so that
        "primes less then N" or "primes with k digits" etc. are not
        permissible) number?

        Richard



        --- In primenumbers@yahoogroups.com, chriscard1@n... wrote:
        > "mad37wriggle" <fitzhughrichard@h...> wrote:
        >
        > >
        > >Are there any types/forms of primes of which there are
        definitely
        > >only a finite number? I'm sure there must be, but can't think of
        > >any off the top of my head. I'd be interested to see the proofs
        of
        > >finitude...
        > >
        > Even primes?
        >
        >
        > Chris
        >
        >
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      • Gary Chaffey
        Fermat Primes are only finite in number. I think I would be right in saying there are infinitely many types of primes which yield only a finite number of
        Message 3 of 14 , Oct 1, 2003
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          Fermat Primes are only finite in number.
          I think I would be right in saying there are
          infinitely many 'types' of primes which yield only a
          finite number of primes
          Gary

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        • Paul Jobling
          ... The heuristics certainly say that, but has it been definitely proven? __________________________________________________ Virus checked by MessageLabs Virus
          Message 4 of 14 , Oct 1, 2003
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            > Fermat Primes are only finite in number.

            The heuristics certainly say that, but has it been definitely proven?

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          • jbrennen
            ... Still, that restriction allows trivial solutions. I came up with this one in just a few minutes, and there are an infinite number of examples like this:
            Message 5 of 14 , Oct 1, 2003
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              --- "mad37wriggle" wrote:
              >
              > OK let me attempt to rephrase that so as to avoid "trivial"
              > solutions...
              >
              > Are there any types/forms of primes of which there are definitely
              > only a finite (>1 and taken over all possible primes so that
              > "primes less then N" or "primes with k digits" etc. are not
              > permissible) number?

              Still, that restriction allows "trivial" solutions. I came up
              with this one in just a few minutes, and there are an infinite
              number of examples like this:


              There are exactly two primes which are of the form:

              4*x^2 - 31*x + 60, with x an integer
            • Andy Swallow
              ... Not true. The number of Fermat primes is suspected to be finite, but it has not been proved. As far as I know, anyway. Unless my books are out of date!
              Message 6 of 14 , Oct 1, 2003
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                > Fermat Primes are only finite in number.

                Not true.
                The number of Fermat primes is suspected to be finite, but it has not
                been proved. As far as I know, anyway. Unless my books are out of date!

                Andy
              • mikeoakes2@aol.com
                In a message dated 01/10/03 15:25:15 GMT Daylight Time, caldwell@utm.edu ... Not the second of these: restrict to x = 2 and you have made a probably-false
                Message 7 of 14 , Oct 1, 2003
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                  In a message dated 01/10/03 15:25:15 GMT Daylight Time, caldwell@...
                  writes:


                  > Primes of the form x^2-1 (or x^n-1)...
                  >
                  Not the second of these: restrict to x = 2 and you have made a probably-false
                  statement:-)

                  Mike



                  [Non-text portions of this message have been removed]
                • Gary Chaffey
                  In an earlier mail I stated:- Fermat Primes are only finite in number. I think I should of worded this more carefully . I know that this is only a conjecture
                  Message 8 of 14 , Oct 1, 2003
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                    In an earlier mail I stated:-
                    Fermat Primes are only finite in number.
                    I think I should of worded this 'more carefully'. I
                    know that this is only a conjecture but it is another
                    one of those conjectures which although hasn't been
                    proven most evidence points this way.



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                  • Gary Chaffey
                    Fermat Primes are only finite in number. P.S. I do have a proof for this but it will not fit in the margin!!!
                    Message 9 of 14 , Oct 1, 2003
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                      Fermat Primes are only finite in number.
                      P.S.
                      I do have a proof for this but it will not fit in the
                      margin!!!




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                    • Paul Jobling
                      On this subject, is there any set of primes that has been shown to have a finite - but unknown - number of elements? I can t think of any, though there are
                      Message 10 of 14 , Oct 1, 2003
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                        On this subject, is there any set of primes that has been shown to have a
                        finite - but unknown - number of elements? I can't think of any, though there
                        are many which are heuristically thought to be finite (such as the Fermat
                        primes).

                        - Paul.


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