Mignotte, Maurice; de Weger, Benjamin M. M. On the diophantine equations \(x^ 2+ 74= y^ 5\) and \(x^ 2+ 86= y^ 5\). (English) Zbl 0847.11011 Glasg. Math. J. 38, No. 1, 77-85 (1996). The authors prove that the only solution in positive integers of \(x^2+ 74= y^5\) is \((x, y)= (13, 3)\) and that \(x^2+ 86= y^5\) is impossible in positive integers. This is a complement to Theorem 1 of J. H. E. Cohn [Acta Arith. 65, 367-381 (1993; Zbl 0795.11016)], in combination with which it proves the following theorem: The equation \(x^2+ C= y^n\) in positive integers \(x\), \(y\), \(n\) with \(n\geq 2\) has only the solutions \(x= 13\), \(y= 3\), \(n= 5\) and \(x= 985\), \(y= 99\), \(n= 3\) when \(C= 74\), and has no solution when \(C= 86\). The authors first reduce their problem to the solution of three quintic Thue equations, which they solve following the lines of the reviewer and the second author [J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)] with the following modifications: (i) They use the recent lower bound of A. Baker and G. Wüstholz [J. Reine Angew. Math. 442, 19-62 (1993; Zbl 0788.11026)] for linear forms in logarithms in place of M. Waldschmidt’s older result, and (ii) They reduce the large upper bound using the LLL algorithm in a considerably more efficient way, following a recent clever idea of Y. Bilu [Isr. J. Math. 90, No. 1-3, 235-252 (1995; Zbl 0840.11028)].In order to get all necessary information (algebraic-number-theoretic “details”) for the relevant quintic fields, the authors used PARI 1.38 and checked their computations with KANT 2. Reviewer: N.Tzanakis (Iraklion) Cited in 2 ReviewsCited in 15 Documents MSC: 11D41 Higher degree equations; Fermat’s equation Keywords:diophantine equations; lower bound for linear forms in logarithms; reduced upper bound; solution in positive integers; quintic Thue equations; quintic fields Citations:Zbl 0795.11016; Zbl 0657.10014; Zbl 0788.11026; Zbl 0840.11028 Software:KANT/KASH PDFBibTeX XMLCite \textit{M. Mignotte} and \textit{B. M. M. de Weger}, Glasg. Math. J. 38, No. 1, 77--85 (1996; Zbl 0847.11011) Full Text: DOI References: [1] DOI: 10.1016/0022-314X(89)90014-0 · Zbl 0657.10014 [2] Baker, J. Reine Angew. Math. 442 pp 19– (1993) · Zbl 0788.11026 [3] Cohn, Acta Arith 65 pp 367– (1993) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.