- No prime p is (p-1)modulo 4, as primes that are(p-1)modulo 2 are

[(p-1)/2]modulo4, and if the order of pmodulo2 is less than p-1, then

the modulo4 of p is either the same as the modulo 2 value or is 1/2 of

the value.

It should be possible therefore to determine values k in the power

series k*4^n+/1, which generate, for all values of n, no factors

smaller than any given prime value through using a modular sieve

process which is much more efficient that choosing the equivalent

primorial or even payam.

As a result, the potential for long Cunningham Chains base 4 becomes

apparent.

So far, the longest I have found (and found after only 3 minutes of

checking!!) is length 13, namely k= 6703351518. A further 5 hours of

checking produced a fair selection of length 10 chains.

Whilst maybe it will be difficult to find one of length 17, it seems

that the trade off between the modular sieve and working in base 4

instead of base 2 might work in favour of the sieve, although I have

no way of proving this. - --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
>

Hmm, must have been drunk when I wrote this. For modulo read modulo

> No prime p is (p-1)modulo 4, as primes that are(p-1)modulo 2 are

> [(p-1)/2]modulo4, and if the order of pmodulo2 is less than p-1, then

> the modulo4 of p is either the same as the modulo 2 value or is 1/2 of

> the value.

>

> It should be possible therefore to determine values k in the power

> series k*4^n+/1, which generate, for all values of n, no factors

> smaller than any given prime value through using a modular sieve

> process which is much more efficient that choosing the equivalent

> primorial or even payam.

>

> As a result, the potential for long Cunningham Chains base 4 becomes

> apparent.

>

> So far, the longest I have found (and found after only 3 minutes of

> checking!!) is length 13, namely k= 6703351518. A further 5 hours of

> checking produced a fair selection of length 10 chains.

>

> Whilst maybe it will be difficult to find one of length 17, it seems

> that the trade off between the modular sieve and working in base 4

> instead of base 2 might work in favour of the sieve, although I have

> no way of proving this.

>

order, or multiplicative order.

No base that is square produces a p-1 multiplicative order, 4 is the

smallest square base, it works just as well with bases 9,16,25...