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Arithmetic properties of certain $t$-regular partitions
Number Theory · Submitted 4 Sep 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$.
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For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. For $t\in \{6,10,14,15,18,20,22,26,27,28\}$, Keith and Zanello conjectured that there are no integers $A>0$ and $B\geq 0$ for which $b_t(An+ B)\equiv 0\pmod 2$ for all $n\geq 0$. We prove that, for any $t\geq 2$ and prime $\ell$, there are infinitely many arithmetic progressions $An+B$ for which $\sum_{n=0}^{\infty}b_t(An+B)q^n\not\equiv0 \pmod{\ell}$. Next, we obtain quantitative estimates for the distributions of $b_{6}(n), b_{10}(n)$ and $b_{14}(n)$ modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and $13$-regular partition functions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. | background 1.00 | 0.21 | 0.02 | asserted 0.76 |
| s1 | Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. | contribution 0.61 | 0.96 | 0.60 | asserted 1.00 |
| s2 | We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. | result 0.60 | 0.96 | 0.18 | asserted 1.00 |
| s3 | We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. | result 0.85 | 0.95 | 0.14 | asserted 1.00 |
| s4 | For $t\in \{6,10,14,15,18,20,22,26,27,28\}$, Keith and Zanello conjectured that there are no integers $A>0$ and $B\geq 0$ for which $b_t(An+ B)\equiv 0\pmod 2$ for all $n\geq 0$. | background 0.99 | 0.04 | 0.04 | speculative 0.80 |
| s5 | We prove that, for any $t\geq 2$ and prime $\ell$, there are infinitely many arithmetic progressions $An+B$ for which $\sum_{n=0}^{\infty}b_t(An+B)q^n\not\equiv0 \pmod{\ell}$. | result 0.95 | 0.96 | 0.24 | asserted 1.00 |
| s6 | Next, we obtain quantitative estimates for the distributions of $b_{6}(n), b_{10}(n)$ and $b_{14}(n)$ modulo 2. | contribution 0.55 | 0.96 | 0.06 | asserted 1.00 |
| s7 | We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and $13$-regular partition functions. | contribution 0.96 | 0.93 | 0.04 | asserted 0.96 |
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Original abstract
For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. For $t\in \{6,10,14,15,18,20,22,26,27,28\}$, Keith and Zanello conjectured that there are no integers $A>0$ and $B\geq 0$ for which $b_t(An+ B)\equiv 0\pmod 2$ for all $n\geq 0$. We prove that, for any $t\geq 2$ and prime $\ell$, there are infinitely many arithmetic progressions $An+B$ for which $\sum_{n=0}^{\infty}b_t(An+B)q^n\not\equiv0 \pmod{\ell}$. Next, we obtain quantitative estimates for the distributions of $b_{6}(n), b_{10}(n)$ and $b_{14}(n)$ modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and $13$-regular partition functions.
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