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## Re: sufficient test for primes with certificate

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• ... I did verify that point and all your other points. See the test: g = 21283; a = 2969; p = 29539; {if(p%4 == 3 && !issquare(a) && kronecker(a,p) == 1 &&
Message 1 of 33 , Jul 16, 2012
"bhelmes_1" <bhelmes@...> wrote:

> You did not verify the point 4.

I did verify that point and all your other points.

See the test:

g = 21283;
a = 2969;
p = 29539;

{if(p%4 == 3 &&
!issquare(a) &&
kronecker(a,p) == 1 &&
Mod(a,p)^((p-1)/2) == 1 &&
Mod((1+x)*Mod(1,p),x^2-a)^((p-1)/2) == g*x &&
Mod(g,p)^2*a == 1 &&
!isprime(p), print(" Counterexample!"));}

In the line

> Mod((1+x)*Mod(1,p),x^2-a)^((p-1)/2) == g*x

I work modulo p and modulo x^2-a.
Here x is /any/ solution to x^2 = a mod p.
(There are in fact, 4 such solutions
since p is composite.)

Now square that line and we get

Mod((1+x)*Mod(1,p),x^2-a)^(p-1) == 1

since (g*x)^2 = g^2*a mod p
and the code tests that

> Mod(g,p)^2*a == 1

Hence this counterexample passes /every/
test that you have imposed. You have
no place left to hide your mistakes.

Best wishes

David
• ... I compiled a better version of my code with gmp 5.0.5, on a different box running at 3.6GHz and got some better timings { 17.505s pfgw (3-prp) 1m1.986s
Message 33 of 33 , Sep 21, 2012
--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
:

> I ran various "minimal \lambda+2" tests on Gilbert Mozzo's 20,000 digit PRP, 5890*10^19996+2^66422-3 (x=1), using a 2.4GHz core:
> {
> 0m32.374s pfgw64 (3-prp)
> 1m9.876s pfgw64 -t
> 1m53.535s pfgw64 -tp
> 3m0.483s pfgw64 -tc
> 5m12.972s pfgw64 scriptify
> 4m4.811s gmp (-O3/no pgo)
> 4m9.148 pari-gp
> 1m15s theoretical Woltman implementation
> }
>

I compiled a better version of my code with gmp 5.0.5, on a different box running at 3.6GHz and got some better timings
{
17.505s pfgw (3-prp)
1m1.986s pfgw -tp
1m13.789s gmp (-O3/no pgo)
}

Paul
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