Sums of two squares and a power

TL;DR

This paper proves the existence of infinitely many positive integers that cannot be expressed as the sum of two squares plus a k-th power, providing new counterexamples to the Hasse principle for all k ≥ 3.

Contribution

It extends previous results by constructing infinite counterexamples to the representation of integers as sums of two squares and a k-th power, with specific congruence conditions.

Findings

Infinite integers not representable as x^2 + y^2 + z^k for k ≥ 3

Counterexamples demonstrate failures of the Hasse principle

Provides new insights into sum of squares and power representations

Abstract

We extend results of Jagy and Kaplansky and the present authors and show that for all $k\geq 3$ there are infinitely many positive integers $n$, which cannot be written as $x^2+y^2+z^k=n$ for positive integers $x,y,z$, where for $k\not\equiv 0 \bmod 4$ a congruence condition is imposed on $z$. These examples are of interest as there is no congruence obstruction itself for the representation of these $n$. This way we provide a new family of counterexamples to the Hasse principle or strong approximation.