A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the ‘‘striking” fact that, if for any $A\equiv \pm 1(\mathrm{mod}6)$ the multipartition function $p_A(n)$ has positive odd density, then so does $p(n)$. Similarly, the positive odd density of any $p_A(n)$ with $A\equiv 3(\mathrm{mod}6)$ would imply that of $p_3(n)$.

Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture.

Parkin-Shanks 的一个著名猜想预测， $磷(n)$密度为 1/2 是奇数。然而，尽管过去几十年做了大量的工作，但即使显示这种密度是积极的，似乎也遥不可及。在 2018 年与 Judge 的一篇论文中，我们引入了一种不同的方法并推测了“惊人”的事实，即如果有任何$\mathrm{一种}\equiv \pm 1(\mathrm{模组}6)$ 多分区函数 ${磷}_{\mathrm{一种}}(n)$ 有正的奇数密度，那么 $磷(n)$. 类似地，任意的正奇数密度${磷}_{\mathrm{一种}}(n)$ 和 $\mathrm{一种}\equiv 3(\mathrm{模组}6)$ 将意味着 ${磷}_3(n)$.

我们的猜想被证明是同一篇论文的早期猜想的推论。在这个简短的说明中，我们提供了一个无条件的证明。一个重要的工具将是陈最近对我们早期猜想的一个特例的突破。