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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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

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

First available in Project Euclid: 3 October 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Class number imaginary quadratic field three squares


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.

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