# PERIODICITY MODULO m AND DIVISIBILITY PROPERTIES OF THE PARTITION FUNCTION(

```@article{Newman1960PERIODICITYMM,
title={PERIODICITY MODULO m AND DIVISIBILITY PROPERTIES OF THE PARTITION FUNCTION(},
author={Morris Newman},
journal={Transactions of the American Mathematical Society},
year={1960},
volume={97},
pages={225-236}
}```
• M. Newman
• Published 1 February 1960
• Mathematics
• Transactions of the American Mathematical Society
has infinitely many solutions in non-negative integers n. This conjecture seems difficult and I have only scattered results. In ?2 of this paper it will be shown that the conjecture is true for m= 5 and 13 by means of congruences derived from the elliptic modular functions, and similar theorems will be proved; for example that p(5n+4)/5 and p(7n+5)/7 fill all residue classes modulo 5 and 7 respectively, infinitely often. In ?1 it will be shown in an elementary way that the conjecture is also…
Distribution of the partition function modulo \$m\$
Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin,
Some Restricted Partition Functions: Congruences Modulo 5
• D. B. Lahiri
• Mathematics
Journal of the Australian Mathematical Society
• 1969
Ramanujan was the first mathematician to discover some of the arithmetical properties of p(n), the number of unrestricted partitions of n. His congruence, for example, is famous [2; 3]. Some progress
ON THE PARITY OF THE PARTITION FUNCTION
Although much is known about the partition function, little is known about its parity. For the polynomials D(x) := (Dx + 1)/24, where D ≡ 23 (mod 24), we show that there are infinitely many m (resp.
Distribution of the partition function modulo composite integers M
seem to be distinguished by the fact that they are exceptionally rare. Recently, Ono [O1] has gone some way towards quantifying the latter assertion. More recently, Ono [O2] has made great progress
Computing the Residue Class of Partition Numbers
In 1919, Ramanujan initiated the study of congruence properties of the integer partition function p(n) by showing that p(5n+ 4) ≡ 0 (mod 5) and p(7n+ 5) ≡ 0 (mod 7) hold for all integers n. These
ARITHMETIC OF THE PARTITION FUNCTION
Here we describe some recent advances that have been made regarding the arithmetic of the unrestricted partition function p(n). A partition of a nonnegative integer n is any nonincreasing sequence of
Note on Partitions Modulo 5
In the paper preceding this one, Parkin and Shanks study the distribution of the values of the unrestricted partition function p(n) modulo 2 and come to the conclusion that there is no apparent
ON THE PARITY OF PARTITION FUNCTIONS
• Mathematics
• 2003
Let S denote a subset of the positive integers, and let pS(n) be the associated partition function, that is, pS(n) denotes the number of partitions of the positive integer n into parts taken from S.

## References

SHOWING 1-10 OF 10 REFERENCES
Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function
• M. Newman
• Mathematics
• 1957
If n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p r(n) as 0. In a recent paper (2) the author established the following congruence: Let r = 4, 6, 8, 10,
Congruences for the coefficients of modular forms and for the coefficients of ()
x = exp 2ri-T, im r > 0, have been given by D. H. Lehmer [1], J. Lehner [2; 3], and A. van Wijngaarden [4]. The moduli for which congruence properties have been determined are products of powers of
Remarks on some modular identities
Introduction. We shall consider a certain class of functions invariant with respect to the substitutions of the congruence subgroup Fo(p) of the modular group r. By specializing these functions, we
Proof of Ramanujan’s partition congruence for the modulus 11³
Presented to the Society, October 30, 1948; received by the editors September 13, 1948 and, in revised form, February 9, 1949. 1 The author is greatly indebted to the referee, who made a very careful
Further Identities and Congruences for the Coefficients of Modular Forms
• M. Newman
• Mathematics