Abstract
The parity of p(n), the ordinary partition function, has been studied for at least a century, yet it still remains something of a mystery. Although much work has been done, the known lower bounds for the number of even and odd values of p(n) for n ≤ N still appear to have a great deal of room for improvement. In this paper, we use classical methods to give a new lower bound for the number of odd values of p(n).
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Acknowledgments
The author would like to thank Bruce Berndt, Ae Ja Yee, and Alexandru Zaharescu for their outstanding work in [2] and [3]. Their two papers are essential reading for anyone genuinely interested in the parity of partition functions, and it was their contributions that inspired the author to revisit the parity of p(n). Also, the author would like to thank the referee for a very thorough and quick treatment of the first version of this paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Eichhorn, D. A New Lower Bound on the Number of Odd Values of the Ordinary Partition Function. Ann. Comb. 13, 297–303 (2009). https://doi.org/10.1007/s00026-009-0030-0
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DOI: https://doi.org/10.1007/s00026-009-0030-0