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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

Author

Listed:
  • Yasutsugu Fujita

    (Nihon University)

  • Maohua Le

    (Lingnan Normal College)

  • Nobuhiro Terai

    (Oita University)

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.

Suggested Citation

  • Yasutsugu Fujita & Maohua Le & Nobuhiro Terai, 2023. "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," Indian Journal of Pure and Applied Mathematics, Springer, vol. 54(4), pages 1145-1157, December.
    • Handle: RePEc:spr:indpam:v:54:y:2023:i:4:d:10.1007_s13226-022-00328-4
    DOI: 10.1007/s13226-022-00328-4
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