  The purpose of this note is to introduce a new approach to the study of one
of the most basic and seemingly intractable problems in partition theory,
namely the conjecture that the partition function $p(n)$ is equidistributed
modulo 2.
  Our main result will relate the densities, say $\delta_t$, of the odd values
of the $t$-multipartition functions $p_t(n)$, for several integers $t$. In
particular, we will show that if $\delta_t>0$ for some $t\in
\{5,7,11,13,17,19,23,25\}$, then (assuming it exists) $\delta_1>0$; that is,
$p(n)$ itself is odd with positive density. Notice that, currently, the best
unconditional result does not even imply that $p(n)$ is odd for $\sqrt{x}$
values of $n\le x$. In general, we conjecture that $\delta_t=1/2$ for all $t$
odd, i.e., that similarly to the case of $p(n)$, all multipartition functions
are in fact equidistributed modulo 2.
  Our arguments will employ a number of algebraic and analytic methods, ranging
from an investigation modulo 2 of some classical Ramanujan identities and
several other eta product results, to a unified approach that studies the
parity of the Fourier coefficients of a broad class of modular form identities
recently introduced by Radu.
