The Diophantine equation $(x+1)^k+(x+2)^k+\cdots+(\ell x)^k=y^n$ revisited

Abstract

Let k, ℓ ≥ 2 be fixed integers and C be an effectively computable constant depending only on k and ℓ. In this paper, we prove that all solutions of the equation (x + 1) k + (x + 2) k + ... + (ℓx) k = y n in integers x, y, n with x, y ≥ 1, n ≥ 2, k = 3 and ℓ ≡ 1 (mod 2) satisfy max{x, y, n} < C. The case when ℓ is even has already been completed by Soydan (Publ. Math. Debrecen 91 (2017), pp. 369-382).

Perfect powers in sum of three fifth powers

“…He proved that if k ≥ 1 and n ≥ 2 are fixed, then (1.1) has only finitely many solutions except for the cases (k, n) ∈ {(1, 2), (3,2), (3,4), (5,2)}. In the same paper Schäffer stated the following conjecture on the integral solution of (1.1).…”

“…Recently, Bennett, Patel and Siksek [8] extended the result of Stroeker for n ≥ 3. Zhang and Bai [44] solved the equation (1.2) for k = 2 and r = x. Bartoli and Soydan [36,4] extended the result of Zhang and Bai [44] for k ≥ 2 and r = lx with l ≥ 2.…”

“…Conjecture 1. [Schäffer, [34]] Let k ≥ 1, n ≥ 2 be integers and (k, n) / ∈ {(1, 2), (3,2), (3,4), (5,2)}. The equation…”