Distribution of the Partition Function Modulo m

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 on the complementary question. Namely, he proves that if M ≥ 5 is prime, then there are in fact infinitely many congruences of the form (1.1). Using the ideas in [O2], Weaver [We] has recorded over 7...

Distribution of the partition function modulo

and he conjectured further such congruences modulo arbitrary powers of 5, 7, and 11. Although the work of A. O. L. Atkin and G. N. Watson settled these conjectures many years ago, the congruences have continued to attract much attention. For example, subsequent works by G. Andrews, A. O. L. Atkin, F. Garvan, D. Kim, D. Stanton, and H. P. F. Swinnerton-Dyer ([An-G], [G], [G-K-S], [At-Sw2]), in t...

Extension of Ramanujan’s Congruences for the Partition Function modulo Powers of 5

Permutation Polynomials modulo m

This paper mainly studies problems about so called “permutation polynomials modulo m”, polynomials with integer coefficients that can induce bijections over Zm = {0, · · · , m−1}. The necessary and sufficient conditions of permutation polynomials are given, and the number of all permutation polynomials of given degree and the number induced bijections are estimated. A method is proposed to dete...

Null Polynomials modulo m

This paper studies so-called “null polynomials modulo m”, i.e., polynomials with integer coefficients that satisfy f(x) ≡ 0 (mod m) for any integer x. The study on null polynomials is helpful to reduce congruences of higher degrees modulo m and to enumerate equivalent polynomial functions modulo m, i.e., functions over Zm = {0, · · · , m − 1} generated by integer polynomials. The most well-know...