GENERALIZED LUCAS PRIMES IN THE FERMAT-EULER EQUATION

Document Type : Original Manuscript

Author

Faculty of Computer Science and Mathematics, University of Kufa, P.O. Box 21, 54001, Al Najaf, Iraq.

10.22044/jas.2024.14045.1790

Abstract

The property of having infinitely many prime numbers award these numbers to have many applications in various fields of sciences. One of the most important applications is their use in the creation of many public key cryptosystems' private keys. Therefore, the main aim of this paper is considering a well known form of primes generated by the Fermat-Euler equation $p=x^2+dy^2$ and studying whether or not this form keeps the property of generating infinitely many primes if the unknowns $x$, $y$ and $p$ are terms in certain binary recurrence sequences called the Lucas sequences of the first kind $\{u_n(a,b)\}$ or the second kind $\{v_n(a,b)\}$. In other words, in this paper we present a technique for investigating the integer solutions $(x,y,p)$ of the equation $p=x^2+dy^2$, where the unknowns are terms in $\{u_n(a,b)\}$ or $\{v_n(a,b)\}$. We also apply this technique for determining the solutions $(x,y,p)=(t_i(a,b),t_j(a,b),t_k(a,b))$ with $1 \leq i \leq j \leq k$, where $t_n(a,b)$ represents the general term $u_n(a,b)$ or $v_n(a,b)$ under certain conditions on the integers $a$ and $b$.

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References

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Journal of Algebraic Systems
Volume 14, Issue 2
April 2026
Pages 317-330

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