On certain prime numbers in Lucas sequences

Athab, Ali S.; Hashim, Hayder R.

doi:10.47974/JIM-1616

Open Access Research Article

On certain prime numbers in Lucas sequences

^*Ali S. AthabCorresponding authoralis.alhatm@student.uokufa.edu.iqFaculty of Computer Science and Mathematics University of KufaNajaf, Iraqhttps://orcid.org/0000-0002-5706-8386View full profile → , Hayder R. Hashimhayderr.almuswi@uokufa.edu.iqFaculty of Computer Science and Mathematics University of KufaNajaf, Iraqhttps://orcid.org/0000-0001-5408-7496View full profile →

* Corresponding author · click or hover a name for details

pp. 1181–1192Vol. 26Issue 6September 2023DOI: 10.47974/JIM-1616XML

Published Online:: 08 Sep 2023
Article type:: Research Article
Language:: EN
Article no.:: JIM-1616
Pages:: 1181–1192

Abstract

In 1964, Shanks conjectured that there are infinitely many primes of the form 1/2(x2 + 1). Therefore, the aim of this paper is to introduce a technique for studying whether or not there are infinitely many prime numbers of the form 1/2(x2 + 1) derived from some Lucas sequences of the first kind {Un(P, Q)} or the second kind {Vn(P, Q)}, where P ≥ 1 and Q = ±1. In addition, as applications we represent the procedure of this technique in case of x is an either integer or Lucas number of the first or the second kind with x ≥ 1 and 1 ≤ P ≤ 20.

Keywords

Lucas sequences Diophantine equation Shanks’ conjecture Prime numbers

Subject Classifications

Primary 11D25Secondary 11B39

Acknowledgements

FRH-P917H73

References

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