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Re: [PrimeNumbers] Finite sets of primes

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  • Ignacio Larrosa Cañestro
    ... Up to 1.25*10^15 there are only two Wieferich primes: 1093 and 3511. Prime p is a Wieferich prime if 2^(p-1) = 1 (mod p^2). But there isn t a prove that
    Message 1 of 9 , 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...

      Up to 1.25*10^15 there are only two Wieferich primes: 1093 and 3511. Prime p
      is a Wieferich prime if 2^(p-1) = 1 (mod p^2).

      But there isn't a prove that there are finitely many Wieferich primes.
      Best regards,

      Ignacio Larrosa Cañestro
      A Coruña (España)
      ilarrosa@...
    • Carl Devore
      ... Gauss proved that it could be constructed.
      Message 2 of 9 , Oct 1, 2003
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        On Wed, 1 Oct 2003, xed wrote:
        > Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
        > using compass and straightedge

        Gauss proved that it could be constructed.
      • Paul Leyland
        ... Not only is there not a proof, heuristic arguments suggest that there should be an infinite number of them. Paul
        Message 3 of 9 , Oct 1, 2003
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          > Up to 1.25*10^15 there are only two Wieferich primes: 1093
          > and 3511. Prime p
          > is a Wieferich prime if 2^(p-1) = 1 (mod p^2).
          >
          > But there isn't a prove that there are finitely many Wieferich primes.

          Not only is there not a proof, heuristic arguments suggest that there should be an infinite number of them.


          Paul
        • Chris Caldwell
          ... An n-gon is constructible (with usual rules) if and only if it is a product of a power of two and distinct Fermat primes. So 17 gets us back to the
          Message 4 of 9 , Oct 1, 2003
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            At 11:26 AM 10/1/2003 -0400, Carl Devore wrote:

            >On Wed, 1 Oct 2003, xed wrote:
            >> Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
            >> using compass and straightedge
            >
            >Gauss proved that it could be constructed.

            At 11:26 AM 10/1/2003 -0400, you wrote:
            >On Wed, 1 Oct 2003, xed wrote:
            >> Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
            >> using compass and straightedge
            >
            >Gauss proved that it could be constructed.

            An n-gon is constructible (with usual rules) if and only if it is a product of a power of two and
            distinct Fermat primes. So 17 gets us back to the Fermats which might be finite... but...

            Chris

            (And indeed I left out 2 with x^n-1. The idea is to take any polynomial that factors,
            it will be prime only if at all but one of the factors is a unit (2 make x-1 a unit). But
            that does give an infinite number of such trivial examples.)
          • Andy Swallow
            ... May be within number theory, but are still somewhat artificial examples. It s all just choosing a particular upper bound, and taking all primes less than
            Message 5 of 9 , Oct 3, 2003
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              > Even within number theory, we can easily define an ubounded number of sets
              > of finite groups of primes. For example: the number of primes < a hailstone
              > number run for a given seed. Or the number of primes less than each perfect
              > number as we successively march upward. Or the number of primes less than
              > the multiple-valued logs of any given imaginary number. Und so weiter.

              May be within number theory, but are still somewhat artificial examples.
              It's all just choosing a particular upper bound, and taking all primes
              less than it. That's nothing particularly special, no matter where the
              upper bound comes from.

              But if looking at finite sets, it does no harm to consider all integers,
              not just primes. And the class number problem provides a good example
              for the original question posed...

              Let D<0 be the discriminant of an imaginary quadratic field K, the
              simplest kind of algebraic number field other than the rationals. It has
              been proved that there are only finitely many D for which the class
              number of K is 1, but it is not known exactly what all these
              discriminants are. At least not as far as I know. In any case, this is
              certainly a highly non-trivial answer to the original question, albeit
              with the conditions relaxed slightly. And it remains within number
              theory too...

              Andy

              PS
              The values of D which give class number 1 are: -1,-2,-3,-7,-11,-19,
              -43,-67 and -163. These values were conjectured by Gauss to be the only
              possibilities. It was 1934 before Heilbronn & Linfoot proved that there
              were only finitely many such D. Numerical evidence suggests fairly
              strongly that the list above is complete. Good old Gauss...
            • Andy Swallow
              ... But of course, pardon me, I was being stupid. Apart from -1, all the numbers in that little list are prime, so it s an ideal answer to the original
              Message 6 of 9 , Oct 3, 2003
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                > But if looking at finite sets, it does no harm to consider all integers,
                > not just primes. And the class number problem provides a good example
                > for the original question posed...
                > The values of D which give class number 1 are: -1,-2,-3,-7,-11,-19,
                > -43,-67 and -163. etc.

                But of course, pardon me, I was being stupid. Apart from -1, all the
                numbers in that little list are prime, so it's an ideal answer to the
                original question.

                Andy

                (and I don't want anybody trying to say that 1 is prime!)
              • mikeoakes2@aol.com
                In a message dated 03/10/03 15:05:25 GMT Daylight Time, ... You must have an old copy of Hardy & Wright :-) Mine (5th edn.) goes on to say (p.217):- Stark
                Message 7 of 9 , Oct 3, 2003
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                  In a message dated 03/10/03 15:05:25 GMT Daylight Time,
                  umistphd2003@... writes:


                  > The values of D which give class number 1 are: -1,-2,-3,-7,-11,-19,
                  > -43,-67 and -163. These values were conjectured by Gauss to be the only
                  > possibilities. It was 1934 before Heilbronn & Linfoot proved that there
                  > were only finitely many such D. Numerical evidence suggests fairly
                  > strongly that the list above is complete. Good old Gauss...
                  >

                  You must have an old copy of Hardy & Wright :-)
                  Mine (5th edn.) goes on to say (p.217):-
                  "Stark (1967) proved that this extra field does not exist."

                  In other words, those are indeed the /only/ simple imaginary quadratic
                  fields.
                  I second your salute to the astounding Gauss!

                  Mike


                  [Non-text portions of this message have been removed]
                • Andy Swallow
                  ... Worryingly, I was working from Cohn s advanced number theory, published in 61. That ll teach me to always trust Hardy & Wright more...! Andy
                  Message 8 of 9 , Oct 3, 2003
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                    > You must have an old copy of Hardy & Wright :-)
                    > Mine (5th edn.) goes on to say (p.217):-
                    > "Stark (1967) proved that this extra field does not exist."

                    Worryingly, I was working from Cohn's advanced number theory, published
                    in '61. That'll teach me to always trust Hardy & Wright more...!

                    Andy
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