## Re: "wriggly" probable primes

Expand Messages
• ... Exercise: Find two pairs of positive integers (x,y) such that 4065702994722252685573484796054334194691713593576645739409115721859519 = 5x^2 + 5xy + y^2
Message 1 of 50 , Sep 8, 2010
"Aldrich" <aldrich617@...> wrote:

> appears twice on A = 5x^2 + 5xy + y^2
> and is therefore composite

Exercise: Find two pairs of positive integers (x,y) such that
4065702994722252685573484796054334194691713593576645739409115721859519
= 5x^2 + 5xy + y^2

Comment: Pari-GP's "qfbsolve" enables a solution in two minutes.
Devotees of "issquare", like Aldrich, may take considerably longer.

David
• ... [4] is meaningless, as it stands. You should write a double mod: 4. (1+x)^p = 1-x mod(x^2-a,p) ... There is no reason whatsoever to believe that [1] to [4]
Message 50 of 50 , Sep 29, 2011
"bhelmes_1" <bhelmes@...> wrote:

> 1. let a jacobi (a, p)=-1
> 2. let a^(p-1)/2 = -1 mod p
> 3. a^6 =/= 1 mod p
> 4. (1+sqrt (a))^p = 1-sqrt (a)

[4] is meaningless, as it stands.
You should write a double mod:

4. (1+x)^p = 1-x mod(x^2-a,p)

> 1. Is it possible that there are other exceptions

There is no reason whatsoever to believe that
[1] to [4] establish that p is prime. Morevoer,
some folk believe that, for every epsilon > 0,
the number of pseudoprimes less than x may
exceed x^(1-epsilon), for /sufficiently/ large x.

> 2....
> there is a cyclic order ...

> 3....
> there is a cyclic order ...

The group of units (Z/nZ)* is /not/ cyclic
if n has at least two distinct odd prime fators.

David
Your message has been successfully submitted and would be delivered to recipients shortly.