## Re: "wriggly" pseudoprime puzzle

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• ... Perhaps. I am using the sequential sledge hammer technique, testing all n and a . The 6-selfridge double-lucas testing now stands at n
Message 1 of 50 , Sep 6, 2010
>
>
>
>
> > use a semiprime p = q*r
> ...
> > I chose (r-1)/(q^2-1) = 2/3
>
> Cf. Paul Underwood's semiprimes:
> http://primes.utm.edu/bios/page.php?id=181
> > 105809903
> > 2499327041
> found when investigating 5-selfridge double-Lucas + Fermat tests.
> Here we have smaller ratios, namely
>
> ratio(p)=local(f=factor(p)[,1]);(f[2]-1)/(f[1]^2-1);
> print([ratio(105809903),ratio(2499327041)]);
> [1/10, 1/36]
>
>
> This observation led me to study a comparable ratio, namely
>
> print([ratio(43334121400711)]);
> [1/33]
>
> and then an even simpler ratio, namely
>
> print([ratio(555826983297468634137017311219)]);
> [2/3]
>
>
> Perhaps Paul might find this observation of use,
> for finding more double-Lucas perjuries?
>

Perhaps. I am using the sequential sledge hammer technique, testing all "n" and "a". The 6-selfridge double-lucas testing now stands at n<1.5*10^7 for "a+-2" and n<2.425*10^7 for "a+-1". The latter requires the test gcd(a*210)==1. At a glance it seems that n/gcd is always 6*N-1. I will get around to verifying this for my searched "n", as I have logged the the cases where gcd(a,n) is required.

It would lighten my CPU load if I just went for such semiprimes as noted by David. My technique will take years to reach n=2^32. I have already ruled out Carmichael numbers below this limit,

Paul
• ... [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
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