# 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} }

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…

## 49 Citations

Distribution of the partition function modulo $m$

- Mathematics
- 2000

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

- MathematicsJournal 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

- Mathematics
- 1995

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

- Mathematics
- 2000

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

- Mathematics
- 2016

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

- Mathematics

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

- Mathematics
- 1967

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.…

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