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arXiv · 1607.02614

Sums of two squares and a power

Rainer Dietmann·Christian Elsholtz
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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.

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Rainer Dietmann, Christian Elsholtz. 2016-07-09. Sums of two squares and a power. https://arxiv.org/abs/1607.02614

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