**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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## Details

### 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*

### Abstract

Primes of the form *n*^{2} + 1 show no deviations contrary to the natural integer structure within the modular ring Z_{4} 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.

### Keywords

- Modular rings
- Primes
- Composites

### AMS Classification

- 11A07
- 11A51
- 11B37

### References

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*Forum Mathematicum*, 20(4), 745–756. - Shanks, D. (1959) A sieve method for factoring numbers of the form
*n*^{2 }+ 1.*Mathematical Tables and other Aids to Computation*(now*Mathematics of Computation*), 13, 78–86. - Leyendekkers, J. V., Shannon, A.G., Rybak, J.M. (2007)
*Pattern Recognition: Modular Rings and Integer Structure*. North Sydney: Raffles KvB Monograph No.9. - Sloane, N.J.A. (1973+)
*The On-Line Encyclopedia of Integer Sequences*. A002496. - Inkeri, K., & Sirkesalo, J. (1959) Factorization of certain numbers of the form
*h*.2+^{n}*k*.*Annales Universitasis Turkuensis, Series Ai*., 388. - Riesel, H. (1994)
*Prime Numbers and Computer Methods for Factorization*. Boston: Birkhäuser.

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## Cite this paper

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

ChicagoLeyendekkers, J. V. and A. G. Shannon “Landau’s Fourth Problem.” Notes on Number Theory and Discrete Mathematics 22, no. 4 (2016): 73-77.

MLALeyendekkers, J. V. and A. G. Shannon, “Landau’s Fourth Problem.” Notes on Number Theory and Discrete Mathematics 22.4 (2016): 73-77. Print.