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.
Dedicated to the memory of Wolfgang Schwarz, with admiration for his broad interests, inside and outside mathematics.
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Dietmann, R., Elsholtz, C. (2016). Sums of Two Squares and a Power. In: Sander, J., Steuding, J., Steuding, R. (eds) From Arithmetic to Zeta-Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-28203-9_7
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DOI: https://doi.org/10.1007/978-3-319-28203-9_7
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