On the Diophantine equation \(x^2 + C = y^n\), for \(C = 2^a3^b17^c\) and \(C = 2^a13^b17^c\) (Q2826346)

In the paper under review the authors find all solutions of the Diophantine equation \(x^2+C=y^n\) with \(n\geq 3\), where \(C\) is a positive integer all whose prime factors are the set \(S\) and \(S\) is one of \(\{2,3,17\},~\{2,13,17\}\). The method is similar to the method used to solve the above equation for other sets \(S\). Namely, one writes \(C=du^2\), where \(d\) is squarefree. Since \(du^2\) is not congruent to \(7\) modulo \(8\), when writing \(x^2+du^2=(x+{\sqrt {-d}} u)(x-{\sqrt {-d}} u)\), the two factors on the right-hand side are coprime in \({\mathbb K}={\mathbb Q}[{\sqrt {-d}}]\). For the choices of \(d\) that these authors have to deal with, the class number of \({\mathbb K}\) is never divisible by primes \(p\geq 5\). Thus, using a standard argument based on the primitive divisor theorem for Lucas-Lehmer sequences, the authors deduce that the given equation has no solution if \(n\) is a multiple of a prime \(p\geq 5\), except when \(n=7\) and \(C=2^{a} 13^b 17^c\). This case, as well as the cases when \(n\) is a multiple of \(3\) or \(4\) are solved by reducing the problem to a search for \(S\)-integer points on a finite set of elliptic curves. The paper contains an essential computational part which was carried out with MAGMA v2.18-6.