THE NUMBER OF INTEGRAL SOLUTIONS TO AN EQUATION INVOLVING SUMS OF RADICALS

Abstract

In this short note, we present a Galois-theoretic proof for the following result. Given an integer k ≥ 2 and fixed positive integers n1, . . . , nk, the number of solutions (x1, . . . , xk, y) ∈ (Z≥0) to the equation (1) is finite. This generalises a problem proposed by the authors and selected for the final round of the Romanian Mathematical Olympiad in 2019. In Theorem 2, we prove an interesting lower bound for the number of such solutions in the particular case when n1 = · · · = nk = n. This lower bound involves the number of divisors function. In the same case, we formulate two conjectures regarding the sequence generated by the number of such solutions. In the first conjecture, we speculate that when k = 2, the sequence takes every positive integer value. The second conjecture concerns an asymptotic of that should hold for general values of k ≥ 2. These are supported by extensive computer calculations. 2010 Mathematics Subject Classification: 11B99, 11A25.