We prove that the equation (x − 3r) 3 + (x − 2r) 3 + (x − r) 3 + x 3 + (x + r) 3 + (x + 2r) 3 + (x + 3r) 3 = y p only has solutions which satisfy xy = 0 for 1 ≤ r ≤ 10 6 and p ≥ 5 prime. This article complements the work on the equations (x − r) 3 + x 3 + (x + r) 3 = y p in [2] and (x − 2r) 3 + (x − r) 3 + x 3 + (x + r) 3 + (x + 2r) 3 = y p in [1]. The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.

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We develop methods to study 2-dimensional 2-adic Galois representations ρ of the absolute Galois group of a number field K, unramified outside a known finite set of primes S of K, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on K and S, we show how to determine the determinant det ρ, whether or not ρ is residually reducible, and further information about the size of the isogeny graph of ρ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for K = Q, and for K imaginary quadratic, ρ being the representation attached to a Bianchi modular form.These results form part of the first author's thesis [2].arXiv:1709.06430v5 [math.NT]