R: [PrimeNumbers] Twin primes Criteria (Correction)
- Many thanks to everyone replied my message.
I did not know before the equivalent criteria
Paul Joblin cited.
My claim that the twin primes' rule holds
was based on the following arguments:
(A) n is prime iff [(n-1)/2]!^2 = +/- 1 mod n
(B) (n,n+2) is a prime pair iff
[(n-1)/2]!^2 = +/- 1 mod n = -/+4 mod (n+2)
(note that the sign on the left side of / is for n=4k-1
and the sign on the rigth side of / is for n=4k+1)
If you agree with (A) and (B), whose proofs are skipped, then
(C) when n=4k+1
considering the identity [(n-3)/2]n - 1 = [(n-5)/2](n+2) + 4
it follows that [(n-3)/2]n - 1 = - 1 mod n = +4 mod (n+2)
hence, by (B), [(n-1)/2]!^2 =[(n-3)/2]n - 1 mod n and mod (n+2)
this means that [(n-1)/2]!^2 =[(n-3)/2]n - 1 mod n(n+2)
which leads to 2[(n-1)/2]!^2 = -(5n+2) mod n(n+2)
(D) when n=4k-1
considering the identity [(n+7)/2]n + 1 = [(n+5)/2](n+2) - 4
and proceeding as in (C)
2[(n-1)/2]!^2 = +(5n+2) mod n(n+2) is obtained at last.
Could anyone verify or not the above reasoning?
I would be very grateful if someone send me a copy of Joseph B. Dence
& Thomas P. Dence's paper where twin prime criteria were proved.
Da: Alan Powell [mailto:powella@...]
Inviato: giovedì 29 marzo 2001 21.00
A: Flavio Torasso
Oggetto: [PrimeNumbers] Twin primes Criteria (Correction)
As Paul Jobling pointed out, your conjecture is equivalent to
the criteria (2) and (3) below. A more thorough investigation
shows that both these equivalent versions are indeed correct.
Mike Oakes appears to be mistaken or I am missing something?
I base this on the following Mathematica snippet:
If[Mod[p,4]==1, r=-1, r=+1];
m=Mod[2((p-1)/2)!^2 - r(5*p+2),p(p+2)];
In 1949 using Wilson's Theorem* P. A. Clement published
a proof that p and p+2 are both prime ("twin primes")
if and only if the following congruence holds:
(1) 4((p-1)! + 1) + p = 0 [mod p(p+2)]
Furthermore in 1995 Joseph B. Dence & Thomas P. Dence
reduced the Clement criterion to the following two criteria:
(2) 2((((p-1)/2)!)^2 + 1) + 5p = 0 [mod p(p+2)]
if and only if p and p+2 are primes and p=4k+1
(3) 2((((p-1)/2)!)^2 - 1) - 5p = 0 [mod p(p+2)]
if and only if p and p+2 are primes and p=4k-1
A similar criterion exists for the primality of p and p+d:
If p>1 and d>1 are both integers, then p and p+d are both
prime if and only if:
( 1 ((-1)^d)d! ) 1 1
(4) (p-1)! ( - + ---------- ) + - + --- is an integer
( p p+d ) p p+d
In addition P. A. Clement's paper proves similar necessary and
sufficient criteria for prime triples p, p+2 and p+6 as well
as prime quadruplets p, p+2, p+6, p+8. If these are of general
interest I can post them to the list.
* Wilson's Theorem:
An integer p>1 is prime if and only if (p-1)! + 1 = 0 [mod p]
At 09:06 AM 3/27/01, torasso.flavio@... wrote:
> I found the following rule concerning twin primes:
> 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