Parity of the partition function
Author:
Ken Ono
Journal:
Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 35-42
MSC (1991):
Primary 05A17; Secondary 11P83
DOI:
https://doi.org/10.1090/S1079-6762-95-01005-5
MathSciNet review:
1336698
Full-text PDF Free Access
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Abstract: Let $p(n)$ denote the number of partitions of a non-negative integer $n$. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers $M$ for which $p(M)$ is odd, as well as infinitely many integers $N$ for which $p(N)$ is even (see Subbarao [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus $t$ when $t=1,2,3,4,5,10,12,16,$ and $40.$ Here we announce that there indeed are infinitely many integers $N$ in every arithmetic progression for which $p(N)$ is even; and that there are infinitely many integers $M$ in every arithmetic progression for which $p(M)$ is odd so long as there is at least one such $M$. In fact if there is such an $M$, then the smallest such $M\leq 10^{10}t^7$. Using these results and a fair bit of machine computation, we have verified the conjecture for every arithmetic progression with modulus $t\leq 100,000$.
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F. Garvan, A simple proof of Watson’s partition congruence for powers of 7, J. Australian Math. Soc. (A) 36 (1984), 316-334.
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M. Hirschhorn, On the residue mod 2 and mod 4 of $p(n)$, Acta Arithmetica 38 (1980), 105-109.
M. Hirschhorn, On the parity of $p(n)$ II, J. Combin. Theory (A) 62 (1993), 128-138. 82d:10025
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M. Hirschhorn and M. Subbarao, On the parity of $p(n)$, Acta Arith. 50 4 (1988), 355-356.
M. Knopp, Modular functions in analytic number theory, Markham, 1970.
N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag, 1984.
O. Kolberg, Note on the parity of the partition function, Math. Scand. 7 (1959), 377-378.
M. Newman, Construction and application of a certain class of modular functions, Proc. London Math. Soc. (3) 7 (1956), 334-350.
M. Newman, Construction and application of a certain class of modular functions II, Proc. London Math. Soc. (3) 9 (1959), 373-387.
T. R. Parkin and D. Shanks, On the distribution of parity in the partition function, Math. Comp. 21 (1967), 466-480.
J.-P. Serre, Divisibilité des coefficients des formes modulaires de poids entier, C.R. Acad. Sci. Paris (A) 279 (1974), 679-682.
J. Sturm, On the congruence of modular forms, Springer Lect. Notes 1240 (1984), Springer-Verlag.
M. Subbarao, Some remarks on the partition function, Amer. Math. Monthly 73 (1966), 851-854.
G.N. Watson, Ramanujan’s Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. vol 179 (1938), 97-128.
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Additional Information
Ken Ono
Affiliation:
address Department of Mathematics, The University of Illinois, Urbana, Illinois 61801
Email:
ono@symcom.math.uiuc.edu
Keywords:
Parity conjecture,
partitions,
modular forms
Received by editor(s):
February 28, 1995
Received by editor(s) in revised form:
May 3, 1995
Additional Notes:
The author is supported by NSF grant DMS-9508976.
Communicated by:
Don Zagier
Article copyright:
© Copyright 1995
American Mathematical Society