The Solution of Fermat’s Two Squares Equation and Its Generalization In Lucas Sequences

Authors

Ali S. Athab, Department of Mathematics, Faculty of Computer Science and Mathematics, University of Kufa, Al Najaf, Iraq.Follow
Hayder R. Hashim, Department of Mathematics, Faculty of Computer Science and Mathematics, University of Kufa, Al Najaf, Iraq.Follow

Abstract

As it is well known, there are an infinite number of primes in special forms such as Fermat's two squares form, p=x^2+y^2 or its generalization, p=x^2+y^4, where the unknowns x, y, and p represent integers. The main goal of this paper is to see if these forms still have an infinite number of solutions when the unknowns are derived from sequences with an infinite number of prime numbers in their terms. This paper focuses on the solutions to these forms where the unknowns represent terms in certain binary linear recurrence sequences known as the Lucas sequences of the first and second types.

Keywords

Diophantine equation, Elliptic curves, Fermat’s two squares Theorem, Lucas sequences, Prime numbers

Article Type

Article

How to Cite this Article

Athab, Ali S. and Hashim, Hayder R. (2024) "The Solution of Fermat’s Two Squares Equation and Its Generalization In Lucas Sequences," Baghdad Science Journal: Vol. 21: Iss. 6, Article 21.
DOI: https://doi.org/10.21123/bsj.2023.8786