Congruence relations for r-colored partitions

2022/06/11 by Robert Dicks, Dicks, Robert
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.05449

openalex publication_date 2022/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℓ ≥ 5 be prime. For the partition function p(n) and 5 ≤ ℓ ≤ 31, Atkin found a number of examples of primes Q ≥ 5 such that there exist congruences of the form p(ℓ Q³ n+β) ≡ 0 \pmodℓ. Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every ℓ. In this paper, for a wide range of c ∈ \mathbbF_ℓ, we prove congruences of the form p(ℓ Q³ n+β₀) ≡ c ⋅ p(ℓ Q n+β₁) \pmodℓ for infinitely many primes Q. For a positive integer r, let p_r(n) be the r-colored partition function. Our methods yield similar congruences for p_r(n). In particular, if r is an odd positive integer for which ℓ > 5r+19 and 2^r+2 \not ≡ 2^{± 1} \pmodℓ, then we show that there are infinitely many congruences of the form p_r(ℓ Q³n+β) ≡ 0 \pmodℓ. Our methods involve the theory of modular Galois representations.

Related

• Congruences like Atkin's for the partition function
• Scarcity of congruences for the partition function
• Theta-type congruences for colored partitions
• Congruences like Atkin's for generalized Frobenius partitions
• Congruences of the partition function