Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Find the answer

Let $n$ be the second smallest integer that can be written as the sum of two positive cubes in two different ways. Compute $n$ .

Your answer

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

Solution

A computer search yields that the second smallest number is 4104 . Indeed, $4104=9^{3}+15^{3}=2^{3}+16^{3}$

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