Deducing the positive odd density of p(n) from that of a multipartition function: An unconditional proof
Document Type
Article
Publication Date
12-2021
Department
Department of Mathematical Sciences
Abstract
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≡±1(mod6) the multipartition function pA(n) has positive odd density, then so does p(n). Similarly, the positive odd density of any pA(n) with A≡3(mod6) would imply that of p3(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.
Publication Title
Journal of Number Theory
Recommended Citation
Zanello, F.
(2021).
Deducing the positive odd density of p(n) from that of a multipartition function: An unconditional proof.
Journal of Number Theory,
229, 277-281.
http://doi.org/10.1016/j.jnt.2021.04.027
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/15078