On a conjecture of Schäffer concerning the equation1k+⋯+xk=yn

Abstract

Abstract. We prove Schäffer's conjecture concerning the solutions of the equation in the title under certain assumptions on x, letting the other variables k, n, y be completely free. We also provide upper bounds for n under more moderate conditions. Finally, we give all solutions of the equation in the title for some concrete values of x. Our results rely on assertions describing the precise exponents of 2 and 3 appearing in the prime factorization of S k (x) and on the explicit solution of polynomial-exponential congruences.

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

“…The purpose of the present paper is to extend the results in [6] for all values of x with x < 25. It is important to mention that for this purpose we need different tools than those used in [6]. The reason is that for the remaining values of x with x < 25 (i.e.…”

“…Theorem 2.1. All solutions of equation (1) in positive integers x, k, y, n with x < 25 and n ≥ 3 are given by (8,3,6,4).…”

“…Recently, Hajdu [6] proved that Schäffer's conjecture holds under certain assumptions made on x, letting all the other parameters free. Among other results, he has proved that the conjecture is true if x ≡ 0, 3 (mod 4) and x < 25.…”

“…The reason is that for the remaining values of x with x < 25 (i.e. those with x ≡ 1, 2 (mod 4)) the methods used in [6] are not applicable. To prove our main theorem, we need to combine sharp upper bounds for linear forms in two logarithms and polynomial-exponential congruences, and we also make use of involved computational facilities.…”

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

“…In 2015, Hajdu [9], proved that Schäffer's conjecture holds under certain assumptions on x, letting all the other parameters free. He also proved that the conjecture is true if x ≡ 0, 3 (mod 4) and x < 25.…”