Two triangular number primality tests and twin prime counting in arithmetic progressions of modulus 8

Werner Hürlimann
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 4, Pages 22—29
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Authors and affiliations

Werner Hürlimann
Swiss Mathematical Society, University of Fribourg
1700 Fribourg, Switzerland

Abstract

Two triangular number based primality tests for numbers in the arithmetic progressions 8n ± 1 are obtained. Their use yield a new Diophantine approach to the existence of an infinite number of twin primes of the form (8n−1, 8n+1).

Keywords

  • Primality test
  • Compositeness test
  • Triangular number
  • Arithmetic progression
  • Diophantine curve of degree two
  • Divisor function
  • Twin prime

AMS Classification

  • 11A51
  • 11B25
  • 11D85

References

  1. Riesel, H. (1985) Prime Numbers and Computer Methods for Factorization (2nd ed. 1994), Birkhäuser, Basel.
  2. Agrawal, M., Kayal, N. & Saxena, N. (2004) PRIMES is in P, Annals Math., 160(2), 781–793.
  3. Schoof, R. (2008) Four primality testing algorithms, In: Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography, Math. Sci. Res. Inst. Publ., Survey in Number Theory, Vol. 44, 101–126, Cambridge University Press, Cambridge.
  4. Mollin, R. A. (2002) A brief history of factoring and primality testing B.C. (before computers), Mathematics Magazine, 75(1), 18–29.
  5. Sloane, N. J. A. (1964) The On-Line Encyclopedia of Integer Sequences, https://oeis.org/
  6. Krätzel, E. (1981) Zahlentheorie, Mathematik für Lehrer, Band 19, VEB Deutscher Verlag für Wissenschaften, Berlin.
  7. Dilcher, K. & Stolarsky, K.B. (2005) A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112, 673–681.
  8. Königsberg, S. R. (2011) Characterizations of prime k-tuples using binomial expressions, Int. Math. Forum, 6(44), 2165–2168.

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Cite this paper

APA

Hürlimann, W. (2015). Two triangular number primality tests and twin prime counting in arithmetic progressions of modulus 8. Notes on Number Theory and Discrete Mathematics, 21(4), 22-29.

Chicago

Hürlimann, Werner. “Two Triangular Number Primality Tests and Twin Prime Counting in Arithmetic Progressions of Modulus 8.” Notes on Number Theory and Discrete Mathematics 21, no. 4 (2015): 22-29.

MLA

Hürlimann, Werner. “Two Triangular Number Primality Tests and Twin Prime Counting in Arithmetic Progressions of Modulus 8.” Notes on Number Theory and Discrete Mathematics 21.4 (2015): 22-29. Print.

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