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,
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  • D. B. Lahiri
  • Mathematics
    Journal of the Australian Mathematical Society
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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
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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
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ON THE PARITY OF PARTITION FUNCTIONS
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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