Infinitely many $n$ such that $p(n)$ is odd/even?

Question

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?

Answer 1

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.

Answer 2

Hint: Look at the pentagonal number theorem and the recurrence relation it yields for $p(n)$, and consider what would happen if $p(n)$ had the same parity for all $n\gt n_0$.