论文标题
一致性关系的R色分区
Congruence relations for r-colored partitions
论文作者
论文摘要
令$ \ ell \ geq 5 $为PRIME。对于分区函数$ p（n）$和$ 5 \ leq \ ell \ leq 31 $，Atkin找到了许多示例的示例$ q \ geq 5 $，因此表格$ p（\ ell q^{3} n+β）\ equiv 0 \ equiv 0 \ pmod pmod {\ ell} $ p（\ ell q^{3} N+β）。 最近，艾格伦（Ahlgren），艾伦（Allen）和唐（Tang）证明，每$ \ ell $都有无限的一致性。 In this paper, for a wide range of $c \in \mathbb{F}_{\ell}$, we prove congruences of the form $p(\ell Q^{3} n+β_{0}) \equiv c \cdot p(\ell Q n+β_{1}) \pmod{\ell}$ for infinitely many primes $Q$.对于正整数$ r $，令$ p_ {r}（n）$为$ r $颜色的分区函数。我们的方法对$ p_ {r}（n）$产生类似的一致性。特别是，如果$ r $是一个奇怪的积极整数，$ \ ell> 5r+19 $和$ 2^{r+2} \ not \ equiv 2^{\ pm 1} \ pmod {\ ell} $，那么我们表明，$ p_ p_ {r} \ el e q^^^^^^^^^^^3}（\ el q^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^3}} \ pmod {\ ell} $。我们的方法涉及模块化Galois表示的理论。
Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+β) \equiv 0 \pmod{\ell}.$ Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every $\ell$. In this paper, for a wide range of $c \in \mathbb{F}_{\ell}$, we prove congruences of the form $p(\ell Q^{3} n+β_{0}) \equiv c \cdot p(\ell Q n+β_{1}) \pmod{\ell}$ for infinitely many primes $Q$. For a positive integer $r$, let $p_{r}(n)$ be the $r$-colored partition function. Our methods yield similar congruences for $p_{r}(n)$. In particular, if $r$ is an odd positive integer for which $\ell > 5r+19$ and $2^{r+2} \not \equiv 2^{\pm 1} \pmod{\ell}$, then we show that there are infinitely many congruences of the form $p_{r}(\ell Q^{3}n+β) \equiv 0 \pmod{\ell}$. Our methods involve the theory of modular Galois representations.
