Proceedings of the Japan Academy, Series A, Mathematical Sciences

The representation function for sums of three squares along arithmetic progressions

Paul Pollack

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Abstract

For positive integers $n$, let $r(n) = \#\{(x,y,z) \in\mathbf{Z}^{3}: x^{2}+y^{2}+z^{2}=n\}$. Let $g$ be a positive integer, and let $A\bmod{M}$ be any congruence class containing a squarefree integer. We show that there are infinitely many squarefree positive integers $n\equiv A\bmod{M}$ for which $g$ divides $r(n)$. This generalizes a result of Cho.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 8 (2016), 96-99.

Dates
First available in Project Euclid: 3 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.pja/1475499415

Digital Object Identifier
doi:10.3792/pjaa.92.96

Mathematical Reviews number (MathSciNet)
MR3554863

Zentralblatt MATH identifier
06673661

Subjects
Primary: 11R29: Class numbers, class groups, discriminants 11R11: Quadratic extensions
Secondary: 11E25: Sums of squares and representations by other particular quadratic forms

Keywords
Class number imaginary quadratic field three squares

Citation

Pollack, Paul. The representation function for sums of three squares along arithmetic progressions. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 8, 96--99. doi:10.3792/pjaa.92.96. https://projecteuclid.org/euclid.pja/1475499415


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