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The equation x 2 n + y 2 n = z 5

Michael A. Bennett (2006)

Journal de Théorie des Nombres de Bordeaux

We show that the Diophantine equation of the title has, for n > 1 , no solution in coprime nonzero integers x , y and z . Our proof relies upon Frey curves and related results on the modularity of Galois representations.

The method of infinite ascent applied on A 4 ± n B 3 = C 2

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

Each of the Diophantine equations A 4 ± n B 3 = C 2 has an infinite number of integral solutions ( A , B , C ) for any positive integer n . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when A , B and C are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions ( A , B , C ) of the Diophantine equation a A 3 + c B 3 = C 2 for any co-prime integer pair ( a , c ) .

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