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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.00004 |
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| _version_ | 1866917940956233728 |
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| author | Du, Julia Q. D. Yao, Olivia X. M. |
| author_facet | Du, Julia Q. D. Yao, Olivia X. M. |
| contents | Recently, Andrews and Paule introduced a partition function $PDN1(N)$ which denotes the number of partition diamonds with $(n+1)$ copies of $n$ where summing the parts at the links gives $N$. They also presented the generating function for $PDN1(n)$ and proved several congruences modulo 5,7,25,49 for $PDN1(n)$. At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical $ p(5^kn +d_k) \equiv 0 \pmod {5^k}$, where $24d_k\equiv 1 \pmod {5^k}$ and $k\geq 1$. In this paper, we give an answer of Andrews and Paule's open problem by proving three congruences modulo arbitrary powers of $5$ for $PDN1(n)$. In addition, we prove two congruences modulo arbitrary powers of $7$ for $PDN1(n)$, which are analogous to Watson's congruences for $p(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Congruences modulo arbitrary powers of $5$ and $7$ for Andrews and Paule's partition diamonds with $(n+1)$ copies of $n$ Du, Julia Q. D. Yao, Olivia X. M. Number Theory Recently, Andrews and Paule introduced a partition function $PDN1(N)$ which denotes the number of partition diamonds with $(n+1)$ copies of $n$ where summing the parts at the links gives $N$. They also presented the generating function for $PDN1(n)$ and proved several congruences modulo 5,7,25,49 for $PDN1(n)$. At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical $ p(5^kn +d_k) \equiv 0 \pmod {5^k}$, where $24d_k\equiv 1 \pmod {5^k}$ and $k\geq 1$. In this paper, we give an answer of Andrews and Paule's open problem by proving three congruences modulo arbitrary powers of $5$ for $PDN1(n)$. In addition, we prove two congruences modulo arbitrary powers of $7$ for $PDN1(n)$, which are analogous to Watson's congruences for $p(n)$. |
| title | Congruences modulo arbitrary powers of $5$ and $7$ for Andrews and Paule's partition diamonds with $(n+1)$ copies of $n$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.00004 |