A note on the solution to the generalized Ramanujan–Nagell equation $$\pmb {x^2+(4c)^y=(c+1)^z}$$ x 2 + ( 4 c ) y = ( c + 1 ) z

Yasutsugu Fujita (), Maohua Le () and Nobuhiro Terai ()
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Yasutsugu Fujita: Nihon University
Maohua Le: Lingnan Normal College
Nobuhiro Terai: Oita University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 4, 1145-1157

Abstract: Abstract Let c be a fixed positive integer with $$c>1$$ c > 1 . Very recently, Terai et al. (Int Math Forum 17:1–10, 2022) conjectured that the equation $$x^2+(4c)^y=(c+1)^z$$ x 2 + ( 4 c ) y = ( c + 1 ) z has only one positive integer solution $$(x,y,z)=(c-1,1,2)$$ ( x , y , z ) = ( c - 1 , 1 , 2 ) , except for $$c \in \{5,7,309\}$$ c ∈ { 5 , 7 , 309 } . In this paper, combining certain known results on Diophantine equations with some elementary methods, we verify that this conjecture is true for several cases.

Keywords: Generalized Ramanujan–Nagell equation; Polynomial–exponential Diophantine equation; 11D61 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00328-4

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