- --- Phil Carmody <thefatphil@...> wrote:
> --- Cletus Emmanuel <cemmanu@...> wrote:

Phil,

> >

> > 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

>

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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http://finance.yahoo.com - --- In primenumbers@y..., Cletus Emmanuel <cemmanu@y...> wrote:
>

Actually, Phil's examples are exactly relevant to the

> --- 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...

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. - --- Cletus Emmanuel <cemmanu@...> wrote:
> --- Phil Carmody <thefatphil@...> wrote:

Tabulate the values of O2(p) (the order of 2 mod p),

> > --- 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...

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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