On the generalized Ramanujan-Nagell equation \(x^2+(2c-1)^m=c^n\) (Q2221035)

There are several results on the generalized Ramanujan-Nagell equation of type \[x^2+b^m=c^n\] where \(b,c\) are given integers \(>1\), \(x,n,m\) are variables.\par \textit{N. Terai} [Bull. Aust. Math. Soc. 90, No. 1, 20--27 (2014; Zbl 1334.11020)] showed that if \(2c-1\) is prime and \(2c-1\equiv 3,5 (\mod 8)\), then the equation \(x^2+(2c-1)^m=c^n\) has only the single solution \((x, m, n) = (c - 1, 1, 2)\) and conjectured that the statement is valid for any \(c\).\par In the present paper the authors show, that if \(2c-1=3p^l\) or \( 5p^l\) with a prime \(p\) and a positive integer \(l\), then the conjecture is true.