TY  - JOUR
T1  - Notes on Sophie Germain Primes
AU  - Akın, Canan
AU  - Baştan, Recep
PY  - 2018
DA  - December
JF  - Turkish Journal of Mathematics and Computer Science
JO  - TJMCS
PB  - Matematikçiler Derneği
WT  - DergiPark
SN  - 2148-1830
SP  - 18
EP  - 21
VL  - 10
LA  - en
AB  - 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.
KW  - Prime number
KW  - Sophie Germain primes
KW  - Safe Primes
CR  - Alkalay-Houlihan C., Sophie Germain and Special Cases of Fermat’s Last Theorem. http://www.math.mcgill.ca/darmon/courses/12-
13/nt/projects/Colleen-Alkalay-Houlihan.pdf. Accessed: 2017-03-20.
CR  - Bishop, S. A., Okagbue, H. I., Adamu, M. O., Olajide, F. A., Sequences of numbers obtained by digit and iterative digit sums of Sophie Germain primes and its variants, Global Journal of Pure and Applied Mathematics, 12(2)(2016), 1473-1480.
CR  - Bucciarelli, L.L., Dworsky N., Sophie Germain: An essay in the history of the theory of elasticity, Vol. 6., Springer Science and Business Media, Netherland, 2012.
CR  - Caldwell, C.K., Prime Pages. The Top Twenty: Sophie Germain. http://primes.utm.edu/top20/page.php?id=2.
CR  - Clement, P. A., Congruences to sets of primes, Am. Math. Mon. 56 (1949), 23-25.
CR  - Daniloff, L.L., The Work of Sophie Germain and Niels Henrik Abel on Fermat’s Last Theorem. MS thesis. 2017.
CR  - Lampret, S., Sieving 2m-prime pairs, Notes on Number Theory and Discrete Mathematics 20 (2014), 54-46.
CR  - Liu, F., On the Sophie Germain prime conjecture, WSEAS Transactions in Math 10, 2 (2011), 421-430.
CR  - Meireles, M., On Sophie Germain primes. Proc. 13th WSEAS Int. Conf. App. Math. (2008), 370-373.
CR  - Ribenboim, P., 13 Lectures on Fermat’s Last Theorem, Springer-Verlag, New York, 1979.
CR  - Ribenboim, P., Fermat’s Last Theorem for Amateurs, Springer-Verlag, New York, 1999.
CR  - Ribenboim, P., The Little Book of Bigger Primes, 2nd ed., Springer-Verlag, New York, 2004.
UR  - https://dergipark.org.tr/en/pub/tjmcs/issue//447243
L1  - https://dergipark.org.tr/en/download/article-file/607843
ER  -