Related papers: Congruences of the partition function

Let $p$ be a prime and ${\mathcal{P}_{p}}$ the set of positive integers which are prime to $p$. We establish the following interesting congruence \[\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}}…

Let $M$ be a positive integer and $p(n)$ be the number of partitions of a positive integer $n$. Newman's Conjecture asserts that for each integer $r$, there are infinitely many positive integers $n$ such that \[ p(n)\equiv r \pmod{M}. \]…

Partitions wherein the even parts appear in two different colours are known as cubic partitions. Recently, Merca introduced and studied the function $A(n)$, which is defined as the difference between the number of cubic partitions of $n$…

Folsom, Kent, and Ono used the theory of modular forms modulo $\ell$ to establish remarkable ``self-similarity'' properties of the partition function and give an overarching explanation of many partition congruences. We generalize their…

In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$…

Beck introduced two partition statistics $NT(r,m,n)$ and $M_{\omega}(r,m,n)$,which denote the total number of parts in the partition of $n$ with rank congruent to $r$ modulo $m$ and the total number of ones in the partition of $n$ with…

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved…

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty $ as…

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…

A partition of a positive integer $n$ is said to be $t$-core if none of its hook lengths are divisible by $t$. Recently, two analogues, $\overline{a}_t(n)$ and $\overline{b}_t(n)$, of the $t$-core partition function, $c_t(n)$, have been…

For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities…

Almost nothing is known about the parity of the partition function $p(n)$, which is conjectured to be random. Despite this expectation, Ono surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4…

Let $B_{l,m}(n)$ denote the number of $(l,m)$-regular bipartitions of $n$. Recently, many authors proved several infinite families of congruences modulo $3$, $5$ and $11$ for $B_{l,m}(n)$. In this paper, using theta function identities to…

The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where…

In this note we will give various exact formulas for functions on integer partitions including the functions $p(n)$ and $p(n,k)$ of the number of partitions of $n$ and the number of such partitions into exactly $k$ parts respectively. For…

We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has…

A new formula for the partition function $p(n)$ is developed. We show that the number of partitions of $n$ can be expressed as the sum of a simple function of the two largest parts of all partitions. Specifically, if $a_1 + >... + a_k = n$…

Recently, Hirschhorn and the first author considered the parity of the function $a(n)$ which counts the number of integer partitions of $n$ wherein each part appears with odd multiplicity. They derived an effective characterization of the…