On the equation 1 + 2 + ⋯ +x=y for fixed x

Abstract

We provide all solutions of the title equation in positive integers x, k, y, n with 1 ≤ x < 25 and n ≥ 3. For these values of the parameters, our result gives an affirmative answer to a related, classical conjecture of Schäffer. In our proofs we combine several tools: Baker's method (in particular, sharp bounds for the linear combinations of logarithms of two algebraic numbers), polynomial-exponential congruences and computational methods.

On the Diophantine equation (x+ 1) + (x+ 2) + ... + (2x) =y

“…The main tools in the proof of this result were the 2-adic valuation of S k (x) and local methods for polynomial-exponential congruences. Recently Bérczes, Hajdu, Miyazaki and Pink [6], provided all solutions of equation (1.1) with 1 ≤ x < 25 and n ≥ 3. In 2013, Zhang and Bai [2], considered the Diophantine equation (1.4) with k = 2.…”