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تسجيل مستخدم جديدCongruences modulo powers of 5 for $k$-colored partitions
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Dazhao Tang
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Let $p_{-k}(n)$ enumerate the number of $k$-colored partitions of $n$. In this paper, we establish some infinite families of congruences modulo 25 for $k$-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type congruences modulo powers of 5 for $p_{-k}(n)$ with $k=2, 6$, and $7$. For example, for all integers $ngeq0$ and $alphageq1$, we prove that begin{align*} p_{-2}left(5^{2alpha-1}n+dfrac{7times5^{2alpha-1}+1}{12}right) &equiv0pmod{5^{alpha}} end{align*} and begin{align*} p_{-2}left(5^{2alpha}n+dfrac{11times5^{2alpha}+1}{12}right) &equiv0pmod{5^{alpha+1}}. end{align*}
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اقرأ أيضاً
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