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R: [PrimeNumbers] Twin primes Criteria (Correction)

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  • torasso.flavio@enel.it
    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
    Message 1 of 1 , Mar 30, 2001
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      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.

      Best regards

      Flavio Torasso

      -----Messaggio originale-----
      Da: Alan Powell [mailto:powella@...]
      Inviato: giovedì 29 marzo 2001 21.00
      A: Flavio Torasso
      Cc: primenumbers@yahoogroups.com
      Oggetto: [PrimeNumbers] Twin primes Criteria (Correction)


      Hi Flavio

      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:

      Do[p=Prime[i];
      If[Mod[p,4]==1, r=-1, r=+1];
      If[PrimeQ[p]&&PrimeQ[p+2],
      m=Mod[2((p-1)/2)!^2 - r(5*p+2),p(p+2)];
      If[m!=0,Print["False"];
      ];
      ];
      ,{i,2,PrimePi[10^4]}];

      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.

      Regards

      Alan Powell

      * 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
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