--- In

primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:

>

> 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.

>

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

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...