Let $p{̅}_o\left(n\right)$ denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function $p{̅}_o\left(n\right)$ have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for $p{̅}_o\left(n\right)$ modulo 3. For example, we prove that for n, α ≥ 0, $p{̅}_o\left(4^{\alpha }(24n+17)\right)\equiv p{̅}_o\left(4^{\alpha }(24n+23)\right)\equiv 0\left(mod3\right)$.

Let $\overline{pp}\left(n\right)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\overline{pp}(3n+2)\equiv 0\left(mod3\right)$. They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for $\overline{pp}\left(n\right)$. Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for $\overline{pp}\left(n\right)$. Furthermore, they also constructed infinite families of congruences for $\overline{pp}\left(n\right)$ modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several...

What should be assumed about the integral polynomials $f₁\left(x\right),...,f_k\left(x\right)$ in order that the solvability of the congruence $f₁\left(x\right)f₂\left(x\right)\cdots f_k\left(x\right)\equiv 0\left(modp\right)$ for sufficiently large primes p implies the solvability of the equation $f₁\left(x\right)f₂\left(x\right)\cdots f_k\left(x\right)=0$ in integers x? We provide some explicit characterizations for the cases when $f_j\left(x\right)$ are binomials or have cyclic splitting fields.

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1}\equiv (p/3)3^{p-1}\left(modp²\right)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1+x+x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is ${\sum }_{k=0}^{p-1}\left(\genfrac{}{}{0pt}{}{p-1}{k}\right)\left(\genfrac{}{}{0pt}{}{2k}{k}\right)({(-1)}^k-{(-3)}^{-k})\equiv (p/3)(3^{p-1}-1)\left(modp³\right)$. In addition, we get some new combinatorial identities.

We systematically investigate the expressions and congruences for both a one-parameter family $\left\{G_n\left(x\right)\right\}$ as well as a two-parameter family $\left\{G_n(r,m)\right\}$ of sequences.

We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences ${\sum }_{i=1}^k{a}_i{x}_i\equiv b(modn)$. In particular, we obtain explicit expressions for the number of solutions, where $x_i$’s are squares modulo $n$. In addition, we obtain expressions for the number of solutions with order restrictions $x_1\ge \cdots \ge x_k$ or with strict order restrictions $x_1>\cdots >x_k$ in some special cases. In these results, the expressions for the number of solutions involve...

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(h{p}^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{k)/m^k\left(modp²\right)}}$, where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^a>3$, then $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(h{p}^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{{k)(-h/2)}^k\equiv ((1-2h)/\left(p^a\right))(1+h({(4-2/h)}^{p-1}-1))\left(modp²\right)}}$, where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^a>3$ then $\genfrac{}{}{0pt}{}{{\sum }_{k=0}^{p^a-1}(p^a-1}{\genfrac{}{}{0pt}{}{k\left)\right(2k}{{k)(-1)}^k\equiv 3^{p-1}(p^a/3)\left(modp²\right)}}$.

For k = 1,2,... let $H_k$ denote the harmonic number ${\sum }_{j=1}^{k}1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have ${\sum }_{k=1}^{p-1}\left(H_k\right)/\left(k{2}^k\right)\equiv 7/24p{B}_{p-3}\left(modp²\right)$, ${\sum }_{k=1}^{p-1}\left(H_{k,2}\right)/\left(k{2}^k\right)\equiv -3/8B_{p-3}\left(modp\right)$, and ${\sum }_{k=1}^{p-1}\left(H{²}_{k,2n}\right)/\left(k^{2n}\right)\equiv (\left(\genfrac{}{}{0pt}{}{6n+1}{2n-1}\right)+n)/(6n+1)p{B}_{p-1-6n}\left(modp²\right)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k,m}:={\sum }_{j=1}^{k}1/\left(j^m\right)$.

The topic of the paper are $\Omega$-algebras, where $\Omega$ is a complete lattice. In this research we deal with congruences and homomorphisms. An $\Omega$-algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an $\Omega$-valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce $\Omega$-valued congruences, corresponding quotient $\Omega$-algebras and $\Omega$-homomorphisms and we investigate connections among these notions....

In this work we prove various cases of the so-called “torsion congruences” between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results...

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\overline{spt}\left(n\right)$, ${\overline{spt}}_1\left(n\right)$, ${\overline{spt}}_2\left(n\right)$, and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions....

For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q²}^3\left(q^{2k}\right)/\left((-q²;q²){²}_k(-q;q){²}_{2k}²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 3 (mod 4), ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q³}\left({(q;q³)}_k{(q²;q³)}_k{q}^{3k}\right)\left({(q⁶;q⁶)}_k²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 2 (mod 3), where $\left[p\right]=1+q+\cdots +q^{p-1}$ and $(a;q)ₙ=(1-a)(1-aq)\cdots (1-a{q}^{n-1})$. We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula. ...

We give several different $q$-analogues of the following two congruences of Z.-W. Sun: $$\sum _{k=0}^{(p^r-1)/2}\frac{1}{8^k}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\equiv \left(\frac{2}{p^r}\right)(mod{p}^2)\text{and}\sum _{k=0}^{(p^r-1)/2}\frac{1}{{16}^k}\left(\genfrac{}{}{0pt}{}{2k}{k}\right)\equiv \left(\frac{3}{p^r}\right)(mod{p}^2),$$ where $p$ is an odd prime, $r$ is a positive integer, and $\left(\frac{m}{n}\right)$ is the Jacobi symbol. The proofs of them require the use of some curious $q$-series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.

A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly $k$ prime factors for $k>1$. Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree $n\le x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^k$ solutions modulo $n$. ...

Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and $U_{n+1}=PUₙ+Q{U}_{n-1}$, $V_{n+1}=PVₙ+Q{V}_{n-1}$ for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x². ...