Consider the number of integer partitions $p(n)$ of $n$ whose (product) generating function reads
$$\sum\_{n\geq0}p(n)\,x^n=\prod\_{k\geq1}\frac1{1-x^k}.$$
There are many congruences for $p(n)$ including those due to Ramanujan: $p(5n+4)\equiv\_50, p(7n+5)\equiv\_70$ and $p(11n+6)\equiv\_{11}0$. Here, I would like to ask:



> 
> **QUESTION.** Is this true? For any prime $q$,
> $$\#\{p(n)\,\,\, \text{mod}\,\,\, q: n\in\mathbb{N}\}=q.$$
> That means, do all modular residues appear? In particular, given a prime $q$, does there always exist an $n$ such that $p(n)\,\,\,\text{mod}\,\,\, q=0$?
> 
> 
> 



Congruence residues of integer partitions
Yes, this is true. In fact something even stronger is known:


For arbitrary positive integers $r,j$, and prime $\ell\geq 5$, there are infinitely many values of $n$ for which
$$p(n)=r\pmod{\ell^j}.$$
This is a special case of a conjecture of Newman and is proved in the paper "[Coefficients of half-integral weight modular forms modulo $\ell^j$](https://doi.org/10.1007/s00208-004-0555-9)" by Scott Ahlgren and Matthew Boylan. I believe it is still unknown whether there are infinitely many $n$ for which $p(n)=0\pmod{3}$.


