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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G. Andrews, *The Theory of Partitions*, Addison-Wesley, 1976.
G. Andrews and F. Garvan, *Dyson’s crank of a partition*, Bull. Am. Math. Soc. **18** (1988), 167-171.
A.O.L. Atkin, *Proof of a conjecture of Ramanujan*, Glasgow Math. J. **8** (1967), 14-32.
F. Garvan, *A simple proof of Watson’s partition congruence for powers of 7*, J. Australian Math. Soc. (A) **36** (1984), 316-334.
F. Garvan, *New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11*, Trans. Am. Math. Soc. **305** (1988), 47-77.
F. Garvan and D. Stanton, *Sieved partition functions and $q-$binomial coefficients*, Math. Comp. **55 191** (1990), 299-311.
F. Garvan and D. Stanton, *Cranks and $t-$cores*, Invent. Math. **101** (1990), 1-17.
B. Gordon and K. Hughes, *Multiplicative properties of $\eta -$products II*, A tribute to Emil Grosswald: Number Theory and related analysis, Cont. Math. **143** (1993), Amer. Math. Soc., 415-430.
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
M. Hirschhorn, *Ramanujan’s partition congruences*, Discrete Math. **131** (1994), 351-355. 82d:10025
M. Hirschhorn and D.C. Hunt, *A simple proof of the Ramanujan conjecture for powers of $5$*, J. Reine Angew. Math. **336** (1981), 1-17.
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