Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Find the answer

Let $n$ be the smallest positive integer such that any positive integer can be expressed as the sum of $n$ integer 2015th powers. Find $n$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

In general, if $k \leq 471600000$ , then any integer can be expressed as the sum of $2^{k}+\left\lfloor\left(\frac{3}{2}\right)^{k}\right\rfloor-2$ integer $k$ th powers. This bound is optimal. The problem asking for the minimum number of $k$ -th powers needed to add to any positive integer is called Waring's problem.

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.