Abstract
We show that if c is a positive integer satisfying \(2c-1=3p^l\ \hbox{or}\ 2c-1=5p^l\) with p prime and l positive integer, then the equation \({x^2 + (2c-1)^m}=c^n\) has only the positive integer solution \((x,m,n)=(c-1,1,2)\) without any congruence condition on a prime p.
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The authors thank the referee for the careful reading and helpful comments.
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The first author is supported by JSPS KAKENHI Grant Number 16K05079, and the second author is supported by JSPS KAKENHI Grant Number 18K03247.
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Fujita, Y., Terai, N. On The Generalized Ramanujan–Nagell Equation \(x^2+(2c-1)^m=c^n\). Acta Math. Hungar. 162, 518–526 (2020). https://doi.org/10.1007/s10474-020-01085-8
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DOI: https://doi.org/10.1007/s10474-020-01085-8