Re: [PrimeNumbers] Twin primes' rule
- In a message dated 27/03/2001 15:12:06 GMT Daylight Time,
> I found the following rule concerning twin primes:I have checked the "only if" part of your iff rule with a short program.
> n and n+2 are both prime iff
> 2 [(n-1)/2]!^2 = \pm (5n+2) mod n(n+2)
> the sign being "+" when n=4k-1, "-" when n=4k+1.
> Is it an interesting or trivial result?
> Are there similar congruences for other prime pairs?
> Thanks for any comments
If (n,n+2) is a prime pair, then your equality mod n(n+2) is satisfied for
the 17 prime pairs from (3,5) up to (239,241).
It fails for (269,271) and (281,283).
It is ok for (311,313) and (347,349).
It fails for (419,421).
It is ok for (431,433) and (461,463).
It fails for all the remaining 633 primes pairs up to (46307,46309), which
was the program's limit.