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Sums of perfect powers lack arithmetic progressions

Source: HMMT Invitational Contest 2016, Problem 5

April 22, 2016
combinatoricsAdditive combinatoricsHMICHMMTarithmetic sequencenumber theory

Problem Statement

Let S={a1,…,an}S = \{a_1, \ldots, a_n \} be a finite set of positive integers of size n≥1n \ge 1, and let TT be the set of all positive integers that can be expressed as sums of perfect powers (including 11) of distinct numbers in SS, meaning T={∑i=1naiei∣e1,e2,…,en≥0}. T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. Show that there is a positive integer NN (only depending on nn) such that TT contains no arithmetic progression of length NN.
Yang Liu
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