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.