Landau’s Fourth problem

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 22, 2016, Number 4, Pages 73—77
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Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney, NSW 2006, Australia

A. G. Shannon
Emeritus Professor, University of Technology Sydney, NSW 2007, Australia
Campion College, PO Box 3052, Toongabbie East, NSW 2146, Australia


Primes of the form n2 + 1 show no deviations contrary to the natural integer structure within the modular ring Z4 and the sum of two squares. Hence primes of this form should occur to infinity with other primes. Trend characteristics of primes and composites were compared graphically.


  • Modular rings
  • Primes
  • Composites

AMS Classification

  • 11A07
  • 11A51
  • 11B37


  1. Hardy, G. H., & Heilbronn, H. (1938) Edmund Landau. Journal of the London Mathematical Society, 13(4), 302–310.
  2. Cioabǎ, S. M., & Ram Murty, M. (2008) Expander graphs and gaps between primes. Forum Mathematicum, 20(4), 745–756.
  3. Shanks, D. (1959) A sieve method for factoring numbers of the form n2 + 1. Mathematical Tables and other Aids to Computation (now Mathematics of Computation), 13, 78–86.
  4. Leyendekkers, J. V., Shannon, A.G., Rybak, J.M. (2007) Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No.9.
  5. Sloane, N.J.A. (1973+) The On-Line Encyclopedia of Integer Sequences. A002496.
  6. Inkeri, K., & Sirkesalo, J. (1959) Factorization of certain numbers of the form h.2nk. Annales Universitasis Turkuensis, Series Ai., 388.
  7. Riesel, H. (1994) Prime Numbers and Computer Methods for Factorization. Boston: Birkhäuser.

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Cite this paper

Leyendekkers, J. V., & Shannon, A. G. (2016). Landau’s Fourth problem, Notes on Number Theory and Discrete Mathematics, 22(4), 73-77.

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