The Generalized Lucas Primes in the Landau’s and Shanks’ Conjectures

Authors

  • Ali Sehen Athab Faculty of Computer Science and Mathematics, University of Kufa, P.O. Box 21, 54001 Al Najaf, Iraq
  • Hayder R. Hashim Faculty of Computer Science and Mathematics, University of Kufa, P.O. Box 21, 54001 Al Najaf, Iraq

DOI:

https://doi.org/10.32996/jmss.2023.4.1.4

Keywords:

Lucas sequences, Diophantine equation, Landau’s conjecture, Shanks’ conjecture, prime numbers.

Abstract

Landau’s conjecture and Shanks’ conjecture state that there are infinitely many prime numbers of the forms x2+1 and x4+1 for some nonzero integer , respectively. In this paper, we present a technique for studying whether or not there are infinitely many prime numbers of the form x2+1 or x4+1 derived from some Lucas sequences of the first kind {Un(P,Q)} (or simply, {Un}) or the second kind {Vn(P,Q)} (or simply, {Vn}) , where P greater or equal to 1 and Q= 1 or -1. Furthermore, as applications we represent the procedure of this technique in case of x is either an integer or a Lucas number of the first or the second kind with x greater or equal to 1 and 1 less or equal to P less or equal to 20.

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Published

2023-03-26

Issue

Vol. 4 No. 1

Section

Research Article

How to Cite

Athab, A. S., & Hashim, H. R. (2023). The Generalized Lucas Primes in the Landau’s and Shanks’ Conjectures. Journal of Mathematics and Statistics Studies, 4(1), 41-57. https://doi.org/10.32996/jmss.2023.4.1.4