Sums of two squares and a power

Rainer Dietmann; Christian Elsholtz

Sums of two squares and a power

Research output: Chapter in Book/Report/Conference proceeding › Other contribution

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 x2 + y2 + zk = n for positive integers x, y, z, where for k≢0mod4k≢0mod4 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.

Original language	English
Title of host publication	From arithmetic to zeta-functions
Publisher	Springer, [Cham]
Pages	103-108
Number of pages	6
Publication status	Published - 31 Dec 2016

Cite this

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title = "Sums of two squares and a power",

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 x2 + y2 + zk = n for positive integers x, y, z, where for k≢0mod4k≢0mod4 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.",

author = "Rainer Dietmann and Christian Elsholtz",

year = "2016",

month = dec,

day = "31",

language = "English",

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publisher = "Springer, [Cham]",

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TY - CHAP

T1 - Sums of two squares and a power

AU - Dietmann, Rainer

AU - Elsholtz, Christian

PY - 2016/12/31

Y1 - 2016/12/31

N2 - 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 x2 + y2 + zk = n for positive integers x, y, z, where for k≢0mod4k≢0mod4 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.

AB - 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 x2 + y2 + zk = n for positive integers x, y, z, where for k≢0mod4k≢0mod4 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.

M3 - Other contribution

SP - 103

EP - 108

BT - From arithmetic to zeta-functions

PB - Springer, [Cham]

ER -