Infinitely many $n$ such that $p(n)$ is odd/even? 
We denote by $p(n)$ the number of partitions of $n$. There are infinitely many integers $m$ such that $p(m)$ is even, and infinitely many integers $n$ such that $p(n)$ is odd.

It might be proved by the Euler's Pentagonal Number Theorem. Could you give me some hints?

The theorem is due to O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 1959 377–378, MR0117213 (22 #7995). The review by Mirsky says that the proof uses the Pentagonal Number Theorem, and is extremely short and simple. 
