On the diophantine equations x² + 74 = y⁵ and x² + 86 = y⁵

Abstract

Abstract. J. H. E. Cohn solved the diophantine equations x 2 + 74 = y" and x 2 + 86 = y", with the condition 5 \ n, by more or less elementary methods. We complete this work by solving these equations for 5 | n, by less elementary methods.

On two Diophantine equations of Ramanujan-Nagell type

“…One aspect of the study of equation (1.2) is to determine the integer solutions (x, k, n). [15], and the remaining ones in the recent paper [7]. Recently, several authors become interested in the case when only the prime factors of D are specified.…”

On the diophantine equation x 2 + 2a · 3b · 11c = y n

“…In [25], Mignotte and de Weger dealt with the cases C = 74 and 86, which had not been covered by Cohn. In both these cases, the only interesting value of the exponent n is n = 5. The remaining cases were finally dealt with by Bugeaud, Mignotte and Siksek in [11].…”

On the diophantine equation x2 + p2k = yn

“…Further, he developed a method by which he found all solutions of the above equation for 77 positive values of D ≤ 100. For D = 74 and D = 86, equation (1) was solved by Mignotte and de Weger [31]. By using the theory of Galois representations and modular forms Bennett and Skinner [8] solved (1) for D = 55 and D = 95.…”