Explicit congruences modulo 2048 for overpartitions

The Ramanujan Journal - Let $${bar{p}}(n)$$ denote the number of overpartitions of n. Recently, a number of congruences modulo powers of 2, 3 and 5 have been discovered. The moduli for these...     

New Ramanujan type congruences modulo 5 for overpartitions

We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle.     

Some congruences modulo 5 and 25 for overpartitions

We present two new Ramanujan-type congruences modulo 5 for overpartitions. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.     

Ramanujan-type congruences for broken 2-diamond partitions modulo 3

The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).     

Ramanujan-type congruences modulo powers of 5 and 7

Let b _? (n) denote the number of ?-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of ?. In particular, they showed that for α, n ≥ 0, b ₂₅ (3^2α+3 n+2 · 3^2α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c _N (n) counts the number of bipartitions (λ₁,λ₂) of n such that each part of λ₂ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, \({c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0\) (mod 5 ^j+1), \({c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0\) (mod 7 ^j+1) and b ₂₅ (3^2α+3 · n+2 · 3^2α+2-1) ≡ 0 (mod 3 · 5^2j-1).     

Ramanujan-type congruences modulo 8 for 7- and 23-core partition functions

Recently, Radu and Sellers proved numerous congruences modulo powers of 2 for \( (2k+1)\)-core partition functions by employing the theory of modular forms. In this paper, employing Ramanujan’s theta function identities, we prove many infinite families of congruences modulo 8 for 7-core partition function. Our results generalize the congruences modulo 8 for 7-core partition function discovered by Radu and Sellers. Furthermore, we present new proofs of congruences modulo 8 for 23-core partition function. These congruences were first proved by Radu and Sellers.     

Ramanujan-type congruences for certain generating functions

For nonnegative integers a, b, the function d _a,b (n) is defined in terms of the q-series $\sum_{n=0}^\infty d_{a,b}(n)q^n=\prod_{n=1}^\infty{(1-q^{ an})^b}/{(1-q^n)}$ . We establish some Ramanujan-type congruences for d _a,b (n) by the theory of modular forms with complex multiplication. As consequences, we generalize the famous Ramanujan congruences for the partition function p(n) modulo 5, 7, and 11.     

Congruences for restricted plane overpartitions modulo 4 and 8

The Ramanujan Journal - In 2009, Corteel, Savelief and Vuleti? generalized the concept of overpartitions to a new object called plane overpartitions. In recent work, the author considered a...     

Combinatorial congruences modulo prime powers

Let be any prime, and let and be nonnegative integers. Let and . We establish the congruence

(motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If is greater than one, and are nonnegative integers with , then

We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a -adic order bound given by the authors in a previous paper can be attained when .

     

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:

     

Ankeny-Artin-Chowla type congruences modulo p 3

The goal of this paper is to use algebraic techniques to compute sums of powers of roots of certain polynomials and derive congruences of Ankeny-Artin-Chowla types modulo p ³.     

A Ramanujan-type congruence modulo 5 for 4-colored generalized Frobenius partitions

In his 1984 AMS Memoir, Andrews introduced the \(k\)-colored generalized Frobenius partition function \(c\phi _k(n)\) which denotes the number of generalized Frobenius partitions of \(n\) with \(k\) colors. Recently, Baruah and Sarmah, Lin, and Sellers established several Ramanujan-type congruences for \(c\phi _4(n)\). In this paper, employing some theta identities due to Ramanujan, the \((p, k)\)-parametrization of theta functions given by Alaca, Alaca, and Williams, and some results of Baruah and Sarmah, we prove that \(c\phi _4(20n+11)\equiv 0\ (\mathrm{mod}\ 5)\).     

Proof of a conjecture on a congruence modulo 243 for overpartitions

Periodica Mathematica Hungarica - Let $$\bar{p}(n)$$ denote the number of overpartitions of n. Recently, numerous congruences modulo powers of 2, 3 and 5 were established regarding $$\bar{p}(n)$$ ....     

Explicit congruences for the partition function modulo every prime

Let p(n) denote the number of unrestricted partitions of the positive integer n, and let m be a prime $\geq 13$. We prove, for k = 1, explicit congruences of the form where $r_{m,k}, \delta_{m,k}$ are integers depending on m and k and $\phi_{m,k}(z)$ are explicitly computable level one holomorphic modular forms of small weight. We also give theoretical and numerical support that the congruences also hold for k > 1. Our main idea is a level reduction result for the modular forms which originated from Atkin-Lehner. From our result, we deduce periodicity properties for the partition function with short periods which improve upon recent results of K. Ono. Received: 24 January 2002     

An unexpected Ramanujan-type congruence modulo 7 for 4-colored generalized Frobenius partitions

In this paper, we present an unexpected Ramanujan-type congruence modulo 7 for \(c\phi _4(n)\), which denotes the number of generalized Frobenius partitions of n with 4 colors. This work extends the recent work of Lin on \(c\phi _4\) modulo 7.     

Cubic and quartic congruences modulo a prime

Let p>3 be a prime, and denote the number of solutions of the congruence . In this paper, using the third-order recurring sequences we determine the values of N_p(x³+a₁x²+a₂x+a₃) and N_p(x⁴+ax²+bx+c), and construct the solutions of the corresponding congruences, where a₁,a₂,a₃,a,b,c are integers.     

Infinitely many congruences modulo 5 for 4-colored Frobenius partitions

In his 1984 AMS Memoir, Andrews introduced the family of functions \(c\phi _k(n),\) which denotes the number of generalized Frobenius partitions of \(n\) into \(k\) colors. Recently, Baruah and Sarmah, Lin, Sellers, and Xia established several Ramanujan-like congruences for \(c\phi _4(n)\) relative to different moduli. In this paper, employing classical results in \(q\)-series, the well-known theta functions of Ramanujan, and elementary generating function manipulations, we prove a characterization of \(c\phi _4(10n+1)\) modulo 5 which leads to an infinite set of Ramanujan-like congruences modulo 5 satisfied by \(c\phi _4.\) This work greatly extends the recent work of Xia on \(c\phi _4\) modulo 5.     

$$M_2$$M2-ranks of overpartitions modulo 6 and 10

The Ramanujan Journal - In this paper, we obtain inequalities on $$M_2$$-ranks of overpartitions modulo 6. Let $$overline{N}_2(s,m,n)$$ be the number of overpartitions of n whose $$M_2$$-rank is...     

Waring type congruences involving factorials modulo a prime

In this paper we prove that any residue class λ modulo a large prime number p can be represented in the form