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Published in:
2016 | OriginalPaper | Chapter
Sums of Two Squares and a Power
Authors : Rainer Dietmann, Christian Elsholtz
Published in: From Arithmetic to Zeta-Functions
Publisher: Springer International Publishing
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Abstract
We extend results of Jagy and Kaplansky and the present authors and show that for all k ≥ 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.