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Notes on Sophie Germain Primes

Year 2018, Volume: 10, 18 - 21, 29.12.2018

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.

References

  • 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.
  • 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.
  • 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.
  • Caldwell, C.K., Prime Pages. The Top Twenty: Sophie Germain. http://primes.utm.edu/top20/page.php?id=2.
  • Clement, P. A., Congruences to sets of primes, Am. Math. Mon. 56 (1949), 23-25.
  • Daniloff, L.L., The Work of Sophie Germain and Niels Henrik Abel on Fermat’s Last Theorem. MS thesis. 2017.
  • Lampret, S., Sieving 2m-prime pairs, Notes on Number Theory and Discrete Mathematics 20 (2014), 54-46.
  • Liu, F., On the Sophie Germain prime conjecture, WSEAS Transactions in Math 10, 2 (2011), 421-430.
  • Meireles, M., On Sophie Germain primes. Proc. 13th WSEAS Int. Conf. App. Math. (2008), 370-373.
  • Ribenboim, P., 13 Lectures on Fermat’s Last Theorem, Springer-Verlag, New York, 1979.
  • Ribenboim, P., Fermat’s Last Theorem for Amateurs, Springer-Verlag, New York, 1999.
  • Ribenboim, P., The Little Book of Bigger Primes, 2nd ed., Springer-Verlag, New York, 2004.
Year 2018, Volume: 10, 18 - 21, 29.12.2018

Abstract

References

  • 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.
  • 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.
  • 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.
  • Caldwell, C.K., Prime Pages. The Top Twenty: Sophie Germain. http://primes.utm.edu/top20/page.php?id=2.
  • Clement, P. A., Congruences to sets of primes, Am. Math. Mon. 56 (1949), 23-25.
  • Daniloff, L.L., The Work of Sophie Germain and Niels Henrik Abel on Fermat’s Last Theorem. MS thesis. 2017.
  • Lampret, S., Sieving 2m-prime pairs, Notes on Number Theory and Discrete Mathematics 20 (2014), 54-46.
  • Liu, F., On the Sophie Germain prime conjecture, WSEAS Transactions in Math 10, 2 (2011), 421-430.
  • Meireles, M., On Sophie Germain primes. Proc. 13th WSEAS Int. Conf. App. Math. (2008), 370-373.
  • Ribenboim, P., 13 Lectures on Fermat’s Last Theorem, Springer-Verlag, New York, 1979.
  • Ribenboim, P., Fermat’s Last Theorem for Amateurs, Springer-Verlag, New York, 1999.
  • Ribenboim, P., The Little Book of Bigger Primes, 2nd ed., Springer-Verlag, New York, 2004.
There are 12 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Recep Baştan This is me

Canan Akın 0000-0002-8922-3272

Publication Date December 29, 2018
Published in Issue Year 2018 Volume: 10

Cite

APA Baştan, R., & Akın, C. (2018). Notes on Sophie Germain Primes. Turkish Journal of Mathematics and Computer Science, 10, 18-21.
AMA Baştan R, Akın C. Notes on Sophie Germain Primes. TJMCS. December 2018;10:18-21.
Chicago Baştan, Recep, and Canan Akın. “Notes on Sophie Germain Primes”. Turkish Journal of Mathematics and Computer Science 10, December (December 2018): 18-21.
EndNote Baştan R, Akın C (December 1, 2018) Notes on Sophie Germain Primes. Turkish Journal of Mathematics and Computer Science 10 18–21.
IEEE R. Baştan and C. Akın, “Notes on Sophie Germain Primes”, TJMCS, vol. 10, pp. 18–21, 2018.
ISNAD Baştan, Recep - Akın, Canan. “Notes on Sophie Germain Primes”. Turkish Journal of Mathematics and Computer Science 10 (December 2018), 18-21.
JAMA Baştan R, Akın C. Notes on Sophie Germain Primes. TJMCS. 2018;10:18–21.
MLA Baştan, Recep and Canan Akın. “Notes on Sophie Germain Primes”. Turkish Journal of Mathematics and Computer Science, vol. 10, 2018, pp. 18-21.
Vancouver Baştan R, Akın C. Notes on Sophie Germain Primes. TJMCS. 2018;10:18-21.