Some congruences modulo 2 and 5 for bipartition with 5-core

Some Curious Congruences modulo Primes

Let n be a positive odd integer and let p > n + 1 be a prime. We mainly derive the following congruence: ∑ 0<i1<···<in<p ( i1 3 ) (−1)i1 i1 · · · in ≡ 0 (mod p).

Some New Super-congruences modulo Prime Powers

Let p > 3 be a prime. We show that p−1 k=0 −1/(p + 1) k p+1 ≡ 0 (mod p 5) and p−1 k=0 1/(p − 1) k p−1 ≡ 0 (mod p 4). For any positive integer m ≡ 0 (mod p), we prove that p−1 k=0 (−1) km p/m − 1 k m ≡ 0 (mod p 4), and p−1 k=1 (−1) km k 2 p/m − 1 k m ≡ 1 p p−1 k=1 1 k (mod p 3) if p > 5. The paper also contains some open conjectures.

Congruences for Andrews ’ Spt - Function modulo Powers of 5 , 7 and 13

Abstract. Congruences are found modulo powers of 5, 7 and 13 for Andrews’ smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan’s partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo for all primes ≥ 5 which were conjectured earlier by the author. We extend Ono’s method to handle the pow...

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

Congruences modulo Prime Powers

Let p be any prime, and let α and n be nonnegative integers. Let r ∈ Z and f (x) ∈ Z[x]. We establish the congruence p deg f k≡r (mod p α) n k (−1) k f k − r p α ≡ 0 mod p ∞ i=α ⌊n/p i ⌋ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If α > 1 and l, s, t are nonnegative integers with s, t < p, then 1 ⌊n/p α−1 ⌋! k≡r (...