On Waring's problem: two squares and three biquadrates

TL;DR

This paper explores the representation of large natural numbers as sums of two squares and three biquadrates, contingent on certain unproven hypotheses, extending understanding of Waring's problem in additive number theory.

Contribution

It demonstrates, assuming the Generalised Riemann Hypothesis and Elliott-Halberstam Conjecture, that all sufficiently large numbers meeting specific congruence conditions can be expressed as such sums.

Findings

All large natural numbers n with specified congruence conditions are sums of 2 squares and 3 biquadrates.

Conditional on major unproven hypotheses, the paper extends classical results in Waring's problem.

Provides a framework for future unconditional results in additive number theory.

Abstract

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott-Halberstam Conjecture, we show that all large natural numbers n with 8 not dividing n, n not congruent to 2 modulo 3, and n not congruent to 14 modulo 16, are the sum of 2 squares and 3 biquadrates.