Notes on Sophie Germain Primes

Authors : Recep BAŞTAN, Canan AKIN

Pages : 18-21

View : 44 | Download : 11

Publication Date : 2018-12-29

Article Type : Conference Paper

Abstract :     An elementary method for eliminating $2m$-prime pairs is given by Lampret  [S. Lampret, Sieving $2m$-prime pairs, Notes on Number Theory and Discrete Mathematics Vol. 20, 2014, No.3, 54-46.], where m is an arbitrary positive integer. 2m-prime pairs are related the twin prime pairs since a $2m$-prime pair is a twin prime pair if $m=1$. Lampret gave a characterization for 6n-prime pairs of the form $insert ignore into journalissuearticles values(6k - 1, 6k + 6n - 1);$. In section 2, the Sophie Germain prime and connected safe prime pairs are referred to as $SG$-$S$-prime pairs in short. By using Lampret`s results, we focus on a characterization to obtain SG-S-prime pairs owing to an eliminating method. Thus it is formed instructions for a sieve as an elementary method to find the $SG$-$S$-prime pairs. Moreover we give an example in which we use our instructions to obtain the SG-S-prime pairs up to $250$.       Wilson`s fundamental theorem in number theory gives a characterization of prime numbers via a congruence. A theorem based on Wilson`s Theorem is formulated by Clement [P. A. Clement, Congruences to sets of primes, Am. Math. Mon. 56, 1949, 23-25]. Clement has a characterization of twin primes $insert ignore into journalissuearticles values(n,n+2);$. In section 3, by a congruence, we give a characterization of Sophie Germain primes in the light of the inspiration of Clement`s theorem.
Keywords : Prime number, Sophie Germain primes, Safe Primes

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