On the numbers that are sums of three cubes

TL;DR

This paper investigates which integers can be expressed as sums of three cubes, providing formulas for their representations and exploring connections to the abc-conjecture.

Contribution

It introduces formulas for counting representations of integers as sums of three cubes with a fixed sum and relates the problem to the abc-conjecture.

Findings

Formulas for the number of representations of n as x^3 + y^3 + z^3 with x + y + z = t

Identification of integers representable as sums of three cubes

Discussion of the relationship between the problem and the abc-conjecture

Abstract

We examine what integers are representable as sums of three cubes. We also provide formulas for the number of representations of $x^3+y^3+z^3=n$ under the condition $x+y+z=t$. Also we show how the problem of three cubes is related to $abc-$conjecture.