Notes on Sophie Germain Primes

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 $(6k - 1, 6k + 6n - 1)$. 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 $(n,n+2)$. 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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