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## Re: [PrimeNumbers] Least Exponent conjecture

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• ... 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
--- 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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• ... 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
--- 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.
• ... 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
--- 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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