For positive integers $n$ and $k$, let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$, by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$. Furthermore, in view of these arithmetic properties of $r_{k}(n)$, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

中文翻译：

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平方和和分区全等

对于正整数$n$和$k$， 让$r_{k}(n)$表示表示的数量$n$作为总和$k$正方形，其中具有不同顺序和不同符号的表示被视为不同的。对于给定的正整数$m$，通过二项式系数的一些性质，我们推导出一些无限的同余族$r_{k}(n)$模数$2^{m}$. 此外，鉴于这些算术性质$r_{k}(n)$，我们为过度分区函数和过度分区对函数建立了许多无限的同余族。