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primes modulo 4

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• 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
Message 1 of 2 , Aug 16, 2007
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
Message 2 of 2 , Aug 17, 2007
--- 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...
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