On Waring's problem: two squares, two cubes and two sixth powers

TL;DR

This paper studies the representation of large integers as sums involving squares, cubes, and sixth powers, showing that the expected formula fails only for a small set of integers.

Contribution

It demonstrates that the anticipated asymptotic formula for such representations fails for at most O((log X)^3) integers up to X.

Findings

The asymptotic formula fails for a very small set of integers.

Most integers conform to the predicted representation count.

The failure set size is bounded by a polylogarithmic function.

Abstract

We investigate the number of representations of a large positive integer as the sum of two squares, two positive integral cubes, and two sixth powers, showing that the anticipated asymptotic formula fails for at most O((log X)^3) positive integers not exceeding X.