On two generalized Ramanujan–Nagell equations

Yasutsugu Fujita (), Maohua Le () and Nobuhiro Terai ()
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Yasutsugu Fujita: Nihon University
Maohua Le: Institute of Mathematics
Nobuhiro Terai: Oita University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 2, 859-872

Abstract: Abstract Let c be a positive integer. Then we conjecture that the equations $$x^2+2(2c)^m=(c^2+2)^n$$ x 2 + 2 ( 2 c ) m = ( c 2 + 2 ) n and $$x^2+2(2c)^m=(2c^2+1)^n$$ x 2 + 2 ( 2 c ) m = ( 2 c 2 + 1 ) n have only the trivial positive integer solution (x, m, n) with explicit exceptional cases. In this paper, we verify that these conjectures are true under certain assumptions on c.

Keywords: Generalized Ramanujan–Nagell equations; Generalized Fermat’s equations; Exponential Diophantine equations; Primitive divisors of Lucas numbers; Zsigmondy’s theorem; 11D61 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00527-7

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