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Sums of Two Squares and a Power

In: From Arithmetic to Zeta-Functions

Author

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  • Rainer Dietmann

    (University of London, Department of Mathematics, Royal Holloway)

  • Christian Elsholtz

    (Technische Universität Graz, Institut für Mathematik und Zahlentheorie)

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.

Suggested Citation

  • Rainer Dietmann & Christian Elsholtz, 2016. "Sums of Two Squares and a Power," Springer Books, in: Jürgen Sander & Jörn Steuding & Rasa Steuding (ed.), From Arithmetic to Zeta-Functions, pages 103-108, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-28203-9_7
    DOI: 10.1007/978-3-319-28203-9_7
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