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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 equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

Let f = ∑ n = 1 ∞ a ( n ) q n ∈ S k + 1 / 2 ( N , χ 0 ) be a nonzero cuspidal Hecke eigenform of weight k + 1 2 and the trivial nebentypus χ 0 , where the Fourier coefficients a ( n ) are real. Bruinier and Kohnen conjectured that the signs of a ( n ) are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies { a ( t n 2 ) } n , where t is a squarefree integer such that a ( t ) ≠ 0 . Let q and d be natural numbers such that ( d , q ) = 1 . In this work, we show that { a ( t n 2 ) } n is equidistributed over any arithmetic progression n ≡ d mod q .

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