Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지

THE GENERALIZATION OF CLEMENT'S THEOREM ON PAIRS OF PRIMES

  • 발행 : 2009.01.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this article, we show a generalization of Clement's theorem on the pair of primes. For any integers n and k, integers n and n + 2k are a pair of primes if and only if 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) whenever (n, (2k)!) = (n + 2k, (2k)!) = 1. Especially, n or n + 2k is a composite number, a pair (n, n + 2k), for which 2k(2k)![(n - 1)! + 1] + ((2k)! - 1)n ${\equiv}$ 0 (mod n(n + 2k)) is called a pair of pseudoprimes for any positive integer k. We have pairs of pseudorimes (n, n + 2k) with $n{\leq}5{\times}10^4$ for each positive integer $k(4{\leq}k{\leq}10)$.

키워드

참고문헌

  1. G.H. Hardy and J.E. Littlewood, Some problems of "Partitio Numerrorum", III: On the expression of a number as a sum of primes, Acta Math., 44 (1923), 1-70. Reprinted in Collected papers of G.H. Hardy, Clarendon Press, Oxford, V. 3(1966), 561-630
  2. Y.Y. Park and H.S. Lee, On the sevral differences between primes, J. Appl. Math. &Computing 13(2003), 37-51 https://doi.org/10.1007/BF02936073
  3. H.S. Lee and Y.Y. Park, On the primes with $p{n+1}−p_n$ = 8 and the sum of their reciprocals, J. Appl. Math. & Computing 22(2006), 441-452 https://doi.org/10.1007/BF02896492
  4. D.M. Burton, Elementary number theory, McGraw-Hill, 1980, 97-98
  5. P. Ribenboim, The new book of prime number records, Springer, 1996, 25-26
  6. P.A. Clement, Congruences fo sets of primes, Am. Math. Mon., 56(1949), 23-25 https://doi.org/10.2307/2305816
  7. Stephen Wolfram, The Mathematica book 4th ed., Wolfram Media/Combridge University Press, 1999