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CRITERIA OF SIMULTANEOUS PRIMALITY DUE TO SMARANDACHE

IP.com Disclosure Number: IPCOM000128828D
Original Publication Date: 1999-Sep-11
Included in the Prior Art Database: 2005-Sep-19
Document File: 2 page(s) / 13K

Publishing Venue

Software Patent Institute

Related People

M.L. Perez: AUTHOR [+3]

Abstract

1) Characterization of twin primes 2) Characterization of a pair of primes 3) Characterization of a triplet of primes 4) Characterization of a quadruple of primes 5) More general 6) Even more general

This text was extracted from a PDF file.
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Page 1 of 2

THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Copyright M.L. Perez, 1999 All Rights Reserved

CRITERIA OF SIMULTANEOUS PRIMALITY DUE TO SMARANDACHE

edited by M.L. Perez

1) Characterization of twin primes:

Let p and p+2 be positive odd integers. Then the following statements are equivalent: a) p and p+2 are both prime; b) (p-1)!(3p+2) + 2p+2 is congruent to 0 (mod p(p+2)); c) (p-1)!(p-2)-2 is congruent to 0 (mod p(p+2)); d) [(p-1)!+1]/p + [(p-1)!2+1]/(p+2) is an integer.

2) Characterization of a pair of primes:

Let p and p+k be positive integers, with (p, p+k) = 1. Then: p and p+k are both prime iff (p- 1)!(p+k) + (p+k-1)!p + 2p+k is congruent to 0 (mod p(p+k)).

3) Characterization of a triplet of primes:

Let p-2, p, p+4 be positive integers, coprime two by two. Then: p-2, p, p+4 are all prime iff (p-1)! + p[(p-3)!+1]/(p-2) + p[(p+3)!+1]/(p+4) is congruent to -1 (mod p).

4) Characterization of a quadruple of primes:

Let p, p+2, p+6, p+8 be positive integers, coprime two by two. Then: p, p+2, p+6, p+8 are all prime iff [(p-1)!+1]/p + [(p-1)!2!+1]/(p+2) + [(p-1)!6!+1]/(p+6) + [(p-1)!8!+1]/(p+8) is an integer.

5) More general:

Let p , p , ..., p be positive integers 1, coprime two by two, and 1 2 n

1 <= k <= p , for all i. Then the following statements areequivalent: i i

a) p , p , ..., p are simultaneously prime; 1 2 n

k n i _________ b) Sigma [(p - k )!(k -1)!-(-1) ] | | p i=1 i i i | | j j different from i

is congruent to 0 (mod p p ...p ); 1 2 n

k n i _________ c) (Sigma [(p - k )!(k -1)!-(-1) ] | | p )/(p ...p ) i=1 i i i | | j s+1 n j different from i

M.L. Perez Page 1 Sep 11,...