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Supercongruences for the Almkvist-Zudilin numbers

Tewodros Amdeberhan, Roberto Tauraso (2016)

Acta Arithmetica

We prove a conjecture on supercongruences for sequences that have come to be known as the Almkvist-Zudilin numbers. Some other (naturally) related family of sequences will be considered in a similar vain.

Superelliptic equations arising from sums of consecutive powers

Michael A. Bennett, Vandita Patel, Samir Siksek (2016)

Acta Arithmetica

Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization ( x - 1 ) k + x k + ( x + 1 ) k = z n (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 2,3,4 using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for k = 5 have x = z = 0, and that there are no solutions for k = 6. The chief innovation...

Sur certaines équations fonctionnelles arithmétiques

Régis de La Bretèche, Gérald Tenenbaum (2000)

Annales de l'institut Fourier

Soit p k le k -ième nombre premier. Une fonction arithmétique complètement additive est définie sur ℕ * par la donnée des f ( p k ) et la formule f ( n ) = ∑ k ≥ 1 f ( p k ) v p k ( n ) ( n ≥ 1 ) , où v p désigne la...

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