On the Diophantine equation \(x^2+d^{2l+1}=y^n\) (Q2882504)

For a squarefree integer \(d > 0\), let \(h(-d)\) denote the class number of the field \({\mathbb Q}(\sqrt{-d})\). The authors find that the equation in the title has precisely 12 solutions \((x,y,n,d,l)\) with \(x\), \(y\) positive and coprime integers, \(l\) non-negative, \(n\geq 3\), and \(2 \leq h(-d) \leq 3\). In case \(d\equiv 7 \pmod 8\), the additional requirement \(y\) odd is made.NEWLINENEWLINEAn extended version of a lemma of \textit{J. H. E. Cohn} [Acta Arith. 109, 205--206 (2003; Zbl 1058.11024)] providing a parameterisation for the solutions of the generalized Ramanujan-Nagell equation \(x^2+D=y^n\) is first proved. In case \(n\geq 5\) is prime, an application of the primitive prime divisors theorem of \textit{Yu. Bilu, G. Hanrot} and \textit{P. M. Voutier} [J. Reine Angew. Math. 539, 75--122 (2001; Zbl 0995.11010)] suffices to conclude that there is no solution under the indicated conditions. For the case \(n=4\), the problem is reduced to solving several ternary equations of signature \((m,m,2)\) for which the modular method works. The complete list of solutions to the equation of interest is derived in this case by invoking some results of \textit{M. A. Bennett} and \textit{C. M. Skinner} [Can. J. Math. 56, 23--54 (2004; Zbl 1053.11025)]. Finally, if \(n=3\), the successful approach depends on the value of \(h(-d)\). In case \(h(-d)=2\), a specific ingredient is Chabauty's method for finding all rational points on several hyperelliptic curves of genus 2. To settle the remaining case, the authors borrow from \textit{M. Mignotte} and \textit{B. M. M. de Weger} [Glasg. Math. J. 38, 77--85 (1996; Zbl 0847.11011)] the idea to reduce the problem to the resolution of several Thue-Mahler equations of degree 3. Local considerations suffice to exclude all but three values for \(d\). The proof is completed by using MAGMA to find all the \(S\)-integral points on some elliptic curves.