For a given integer $m$ and any residue $a \pmod{ m}$ that can be written as a sum of 3 squares modulo $m$, we show the existence of infinitely many integers $n \equiv a \pmod{ m}$ such that the number of representations of $n$ as a sum of three squares, $r_3(n)$, satisfies $r_3(n) \gg_m \sqrt{ n} \log \log n$. Consequently, we establish that there are infinitely many integers $n \equiv a \pmod{ m}$ for which the Hurwitz class number $H(n)$ also satisfies $H(n) \gg_m \sqrt{ n} \log \log n$.
Categories: math.NT