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Representation as sums of 33-rd powers

Source: Bulgarian Spring Tournament 2024 12.3

March 31, 2024
number theory

Problem Statement

For a positive integer nn, denote with b(n)b(n) the smallest positive integer kk, such that there exist integers a1,a2,…,aka_1, a_2, \ldots, a_k, satisfying n=a133+a233+…+ak33n=a_1^{33}+a_2^{33}+\ldots+a_k^{33}. Determine whether the set of positive integers nn is finite or infinite, which satisfy:
a) b(n)=12;b(n)=12; b) b(n)=121212.b(n)=12^{12^{12}}.
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