Sums of Two Squares and a Power

Rainer Dietmann () and Christian Elsholtz ()
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Rainer Dietmann: University of London, Department of Mathematics, Royal Holloway
Christian Elsholtz: Technische Universität Graz, Institut für Mathematik und Zahlentheorie

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 103-108 from Springer

Abstract: 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.

Keywords: Hasse principle; Strong approximation; Ternary additive problems; Waring type problems; Primary 11E25; Secondary 11P05 (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-28203-9_7

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DOI: 10.1007/978-3-319-28203-9_7

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