Loading ...
Sorry, an error occurred while loading the content.

Finite sets of primes

Expand Messages
  • xed
    ... Aside from wholly trivial cases which Chris Caldwell has partly covered (even primes, n-digit primes, primes a
    Message 1 of 9 , Oct 1, 2003
    • 0 Attachment
      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...

      Aside from wholly trivial cases which Chris Caldwell has partly covered
      (even primes, n-digit primes, primes a < x < b where x is prime, yadda yadda
      yadda), one particularly obvious yet non-trivial answer to this query would
      involve dragging in boundary conditions from disciplines other than number
      theory. For example: group theory. As we all know, Galois proved that
      algebraic equations > degree 5 have roots which cannot be found by
      Cardano-type cubic or quadratic closed formulas. So all primes < 5 fall into
      a special class for that reason.
      Geometric: Gauss proved that a regular 17-gon cannot be constructed solely
      using compass and straightedge, so all primes < 17 also fall into another
      special class. The same idea could doubtless be carried out by keelhauling
      a wide variety of disciplines into forcible contact with the primes. Viz.,
      algebraic topology: 4-manifolds are known to be peculiarly difficult to find
      minimal surfaces in, if memory serves; physics: a real treasure trove of
      arbitrarily large but finite sets, which would include subsets of primes --
      close packing of various regular solids offer only limited solutions
      including subsets of primes; solid geometry: the Gauss formula for vertices,
      etc.
      By dragging in other disciplines, an unbounded number of sets of finite
      primes could doubtless be classified. The trick involves finding boundary
      conditions unrelated to pure number theory, but calling on different realms
      of mathematics such as plane gemoetry, statistical mechanics, topology, ad
      infinitum.
      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.
    • 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 2 of 9 , Oct 1, 2003
      • 0 Attachment
        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 3 of 9 , Oct 1, 2003
        • 0 Attachment
          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 4 of 9 , Oct 1, 2003
          • 0 Attachment
            > 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 5 of 9 , Oct 1, 2003
            • 0 Attachment
              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 6 of 9 , Oct 3, 2003
              • 0 Attachment
                > 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 7 of 9 , Oct 3, 2003
                • 0 Attachment
                  > 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 8 of 9 , Oct 3, 2003
                  • 0 Attachment
                    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 9 of 9 , Oct 3, 2003
                    • 0 Attachment
                      > 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
                    Your message has been successfully submitted and would be delivered to recipients shortly.