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Re: [PrimeNumbers] Least Exponent conjecture

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  • Cletus Emmanuel
    ... Phil, p is a prime congruent to +/- 1 mod 8. My question now is, Is the order of 2 mod p equal to (p-1)/2? I think so, but have no proof.... The
    Message 1 of 7 , Aug 29, 2002
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      --- Phil Carmody <thefatphil@...> wrote:
      > --- Cletus Emmanuel <cemmanu@...> wrote:
      > >
      > > Phil,
      > > is it safe to say that the order of 2 mod p
      > (prime congruent to +/- 1 mod 8) is (p-1)/2? If
      > > the quadratic equation X^2 - 2 == 0 mod p is
      > solvable, then (2/p) = 1 (Legendre Symbol) and p ==
      > > +/- 1 mod 8.
      > > If you notice, Carol and Kynea numbers are of the
      > form X^2 - 2, where X = 2^k-1 for Carol and X
      > > = 2^k+1 for Kynea...
      >
      > Nope, it divides (p-1)/2. e.g. 5|15, 9|36, 11|44,
      > 7|63, 28|56, 15|75, etc.
      >
      > Phil
      >

      Phil,
      p is a prime congruent to +/- 1 mod 8. My question
      now is, "Is the order of 2 mod p equal to (p-1)/2? I
      think so, but have no proof.... The exasmples you
      gave had nothing to do with the questions I asked...

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    • jbrennen
      ... Actually, Phil s examples are exactly relevant to the question you asked. If p==31, the order of 2 mod p is 5, which divides 15 == (p-1)/2. If p==73, the
      Message 2 of 7 , Aug 29, 2002
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        --- In primenumbers@y..., Cletus Emmanuel <cemmanu@y...> wrote:
        >
        > --- Phil Carmody <thefatphil@y...> wrote:
        > > --- Cletus Emmanuel <cemmanu@y...> wrote:
        > > >
        > > > Phil,
        > > > is it safe to say that the order of 2 mod p
        > > (prime congruent to +/- 1 mod 8) is (p-1)/2?
        > >
        > > Nope, it divides (p-1)/2. e.g. 5|15, 9|36, 11|44,
        > > 7|63, 28|56, 15|75, etc.
        >
        > Phil,
        > p is a prime congruent to +/- 1 mod 8. My question
        > now is, "Is the order of 2 mod p equal to (p-1)/2? I
        > think so, but have no proof.... The exasmples you
        > gave had nothing to do with the questions I asked...

        Actually, Phil's examples are exactly relevant to the
        question you asked.

        If p==31, the order of 2 mod p is 5, which divides 15 == (p-1)/2.
        If p==73, the order of 2 mod p is 9, which divides 36 == (p-1)/2.
        If p==89, the order of 2 mod p is 11, which divides 44 == (p-1)/2.
        If p==127, the order of 2 mod p is 7, which divides 63 == (p-1)/2.
        If p==113, the order of 2 mod p is 28, which divides 56 == (p-1)/2.
        If p==151, the order of 2 mod p is 15, which divides 75 == (p-1)/2.

        Those values of p are all primes congruent to +/-1 mod 8.
      • Phil Carmody
        ... Tabulate the values of O2(p) (the order of 2 mod p), and (p-1)/2 for p in {31, 73, 89, 127, 113, 151 } I ll give you a clue - the first two entries is P :
        Message 3 of 7 , Aug 29, 2002
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          --- Cletus Emmanuel <cemmanu@...> wrote:
          > --- Phil Carmody <thefatphil@...> wrote:
          > > --- Cletus Emmanuel <cemmanu@...> wrote:
          > > >
          > > > Phil,
          > > > is it safe to say that the order of 2 mod p
          > > (prime congruent to +/- 1 mod 8) is (p-1)/2? If
          > > > the quadratic equation X^2 - 2 == 0 mod p is
          > > solvable, then (2/p) = 1 (Legendre Symbol) and p ==
          > > > +/- 1 mod 8.
          > > > If you notice, Carol and Kynea numbers are of the
          > > form X^2 - 2, where X = 2^k-1 for Carol and X
          > > > = 2^k+1 for Kynea...
          > >
          > > Nope, it divides (p-1)/2. e.g. 5|15, 9|36, 11|44,
          > > 7|63, 28|56, 15|75, etc.
          > >
          > > Phil
          > >
          >
          > Phil,
          > p is a prime congruent to +/- 1 mod 8. My question
          > now is, "Is the order of 2 mod p equal to (p-1)/2? I
          > think so, but have no proof.... The exasmples you
          > gave had nothing to do with the questions I asked...

          Tabulate the values of O2(p) (the order of 2 mod p),
          and (p-1)/2 for p in {31, 73, 89, 127, 113, 151 }

          I'll give you a clue - the first two entries is

          P : O2(p) | (p-1)/2
          ---+-------+--------
          31 : 5 | 15
          73 : 9 | 36


          "nothing"?

          Phil



          =====
          "The hottest places in Hell are reserved for those who, in
          times of moral crisis, preserved their neutrality."
          -- John F. Kennedy, 24 June 1963, claiming to quote Dante,
          to whom this has been incorrectly attributed ever since.

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