On a character-twisted analogue of Schäffer's equation

2026/07/16 by Kálmán Györy, Vandita Patel, Ákos Pintér +1
#math.NT

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Abstract

Let f be a positive integer, and let χ be a primitive quadratic character of conductor f. Let k be a positive integer, and write B_k(χ,X) for the k-th Bernoulli polynomial corresponding to χ. Suppose B_k(χ,X) is irreducible and of degree at least 2. Then for 100% of positive integers m divisible by f, the Diophantine equation χ(1) ⋅ (x+1)^k+χ(2) ⋅ (x+2)^k+⋯+χ(m) ⋅ (x+m)^k = yⁿ, has no solutions with x, y, n integers, and n ≥ 2.

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