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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 [+2]

## 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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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

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,...