Re: Unified test for Sophie Germain primes
- --- In firstname.lastname@example.org,
"j_chrtn" <j_chrtn@...> wrote:
> Do you know if this test is something well known?It's not well known because it's quite unnecessary.
The foolproof "unified Sophie" test that works in
*all* cases is much simpler.
If p is prime, then q = 2*p+1 is prime iff 4^p = 1 mod q.
Proof: Simply use base 2 in Pocklington's theorem and
observe that 2^2-1 is coprime to 2*p+1 for every prime p.
Note that this test detects *every* Sophie pair,
including [2,5] and [3,7]. Here is a sanity check:
\\ The rest is silence, signifying consent
Entia non sunt multiplicanda praeter necessitatem :-)
David [second attempt at a reply; apologies if it appears twice]
>--- In email@example.com, "djbroadhurst" <d.broadhurst@...> wrote:Unnecessary but at least correct.
> It's not well known because it's quite unnecessary.
>This is closed to Henri Lifchitz's test 3^p = 1 (mod q) but your's also handles p=2 and p=3.
> If p is prime, then q = 2*p+1 is prime iff 4^p = 1 mod q.
>"There Is More Than One Way To Do It."
> Entia non sunt multiplicanda praeter necessitatem :-)
However, some ways are more efficient than others ;-)