On the solutions of the equation p=x^2+y^2+1 in Lucas sequences
DOI:
https://doi.org/10.31185/wjps.138Keywords:
Lucas sequences, Diophantine equation, Prime numbers.Abstract
In 1970, Motohashi proved that there are an infinite number of primes having the form p=x^2+y^2+1 for some nonzero integers x and y. In this paper, we present a technique for studying the solutions of the equation p=x^2+y^2+1, where the unknowns are derived from some Lucas sequences of the first kind {Un(P, Q)} or the second kind {Vn(P, Q)} with P and Q are certain nonzero relatively primes integers. As applications to this technique, we apply our procedure in case of (x, y, p) = (Ui(P, Q), Uj(P, Q), Uk(P, Q)) or (Vi(P, Q), Vj(P, Q), Vk(P, Q)) with i, j, k > 1, -2 < p < 3 and Q =1 or -1.
References
A. S. Athab and H. R. Hashim. On certain prime numbers in Lucas sequences. Accepted for publication in Journal of Interdisciplinary
Mathematics, pages 1–10, 2023.
A. S. Athab and H. R. Hashim. The generalized Lucas primes in the Landau’s and Shanks’conjectures. Journal of Mathematics and
Statistics Studies, 4(1):41–57, 2023.
H. R. Hashim, L. Szalay, and Sz. Tengely. Markoff-Rosenberger triples and generalized Lucas sequences. Period. Math. Hung.,
(1):188–202, 2022.
E. Landau. Gel¨oste und ungel¨oste Probleme aus der Theorie der Pri zahlverteilung und der Riemannschen Zetafunktion. Proc. 5.
Intern. Math. Congr. 1, 93-108 (1913)., 1913.
J. Leyendekkers and A. Shannon. Pell and Lucas primes. Notes on Number Theory and Discrete Mathematics, 21(3):64–69, 2015.
Y. Motohashi. On the distribution of prime numbers which are of the form x^2+y^2+1. Acta Arith., 16:351–363, 1970.
P. Ribenboim. My numbers, my friends. Popular lectures on number theory. New York, NY:Springer, 2000.
D. Shanks. On numbers of the form n^4+1. Math. Comput., 15:186–189, 1961.
D. Shanks. An analytic criterion for the existence of infinitely many primes of the form 1/2 (n^2+1). Ill. J. Math., 8:377–379, 1964.
L. Somer and M. Kˇr´ıˇz. On primes in Lucas sequences. Fibonacci Q., 53(1):2–23, 2015.
W. A. Stein et al. Sage Mathematics Software (Version 9.0). The Sage Development Team, 2020. http://www.sagemath.org.
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Copyright (c) 2023 Ali Sehen Athab, HAYDER R. HASHIM
This work is licensed under a Creative Commons Attribution 4.0 International License.
