- --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>

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.

>

>

> --- In primenumbers@yahoogroups.com,

> "djbroadhurst" <d.broadhurst@> wrote:

>

> > 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]

>

> in http://tech.groups.yahoo.com/group/primenumbers/message/21673

>

> This observation led me to study a comparable ratio, namely

>

> print([ratio(43334121400711)]);

> [1/33]

>

> in http://tech.groups.yahoo.com/group/primenumbers/message/21788

> and then an even simpler ratio, namely

>

> print([ratio(555826983297468634137017311219)]);

> [2/3]

>

> in http://tech.groups.yahoo.com/group/primenumbers/message/21792

>

> Perhaps Paul might find this observation of use,

> for finding more double-Lucas perjuries?

>

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 - --- In primenumbers@yahoogroups.com,

"bhelmes_1" <bhelmes@...> wrote:

> 1. let a jacobi (a, p)=-1

[4] is meaningless, as it stands.

> 2. let a^(p-1)/2 = -1 mod p

> 3. a^6 =/= 1 mod p

> 4. (1+sqrt (a))^p = 1-sqrt (a)

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....

The group of units (Z/nZ)* is /not/ cyclic

> there is a cyclic order ...

> 3....

> there is a cyclic order ...

if n has at least two distinct odd prime fators.

David