In number theory, the partition function $p(n)$ represents the number of possible partitions of a non-negative integer $n$. For instance, $p(4) = 5$ because the integer $4$ has the five partitions $1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2$ and $4$. (Wikipedia)
Partition function has amazing properties, among which the most beautiful one is Ramanujan's congruences
$$p(5n+4)\equiv0\ (\text{mod}\ 5),\\p(7n+5)\equiv0\ (\text{mod}\ 7),\\p(11n+6)\equiv0\ (\text{mod}\ 11).$$
There are many problems remain open and I want to do some research at some of them. I list some open problems as follows:
For each positive number $n$, the polynomials $\alpha_n(q),\beta_n(q)$ and $\gamma_n(q)$ defined by
$$\prod_{j=1}^\infty(1-q^{3j-2})^2(1-q^{3j-1})^2=\alpha_n(q^3)-q\beta_n(q^3)-q^2\gamma_n(q^3),$$
each has non-negative coefficients.
For each positive number $n$, the polynomials $\nu_n(q),\phi_n(q),\chi_n(q),\psi_n(q)$ and $\omega_n(q)$ defined by
$$\prod_{j=1}^\infty(1-q^{5j-4})(1-q^{5j-3})(1-q^{5j-2})(1-q^{5j-1}) \\=\nu_n(q^5)-q\phi_n(q^5)-q^2\chi_n(q^5)-q^3\psi_n(q^5)-q^4\omega_n(q^5),$$
each has non-negative coefficients.
Remark. The First Borwein Conjecture is proved recently. See here.
- Prime Partition Numbers
It is conjectured that $p(n)$ is prime for infinitely many $n$. However, I think this problems is much harder than the $n^2+1$ conjecture.
Remark. It is easy to derive from Ono's result that every prime divides at least one value of $p(n)$.
- Parity of Partition Function
Prove that
$$\lim_{n\to\infty}\frac{\#\{m\leq n\,|\,p(m)\text{ is even}\}}{n}=\frac12.$$
- Newman's Conjecture
For each integer $r$ and $m$ with $0\leq r<m$, there are infinitely many $n$ such that $p(n)\equiv r\ (\text{mod}\ m)$. If possible, find the density of each residue modulo $m$.
Remark. The conjecture is proved for all prime number $m\geq5$. See here. Moreover, there are useful criteria to attack this conjecture. See here and here.
I want to know more unsolved problems for partition function. And I hope this post will become a reference post.
Any supplement of other open problems is appreciated.