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Let [Formula: see text] be the number of [Formula: see text]-colored partitions of [Formula: see text]. Recently, Keith proved that for [Formula: see text], if [Formula: see text] for all [Formula: see text], then [Formula: see text] is large. We prove that such [Formula: see text] do not exist. Furthermore, for any positive integers [Formula: see text] with [Formula: see text], there exist infinitely many positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text].

Recently the first author proved a congruence proposed in 2006 by Adamchuk: [Formula: see text] for any prime [Formula: see text]. In this paper, we provide more examples (with proofs) of congruences of the same kind [Formula: see text] where [Formula: see text] is a prime such that [Formula: see text], [Formula: see text] is a fraction in [Formula: see text] and [Formula: see text] is a [Formula: see text]-adic integer. The key ingredients are the [Formula: see text]-adic Gamma function [Formula: see text] and a special class of computer-discovered hypergeometric identities.

In 2012, Chan posed six conjectures on congruences for Appell–Lerch sums and five of them were proved by mathematicians. In this paper, we confirm the remaining conjectural congruence of Chan by utilizing an identity due to Hickerson and Mortenson and the theory of modular forms.

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson’s rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of [Formula: see text]. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here.

In this paper, we consider the congruence equation [Formula: see text] with [Formula: see text] where [Formula: see text] denotes the multiplicative inverse of [Formula: see text]. We prove that its number of solutions is asymptotic to [Formula: see text] when [Formula: see text] by estimating a certain average of Kloosterman sums via Gauss sums. On the other hand, when [Formula: see text], the number of solutions has order of magnitude at least [Formula: see text]. It would be interesting to better understand its transition of behavior. By transforming the question slightly, one can relate the problem to a certain first moment of Dirichlet [Formula: see text]-functions at [Formula: see text].

Let [Formula: see text] be a positive odd integer and [Formula: see text] a positive integer with [Formula: see text]. We prove that [Formula: see text] Here [Formula: see text], [Formula: see text] denotes the Jacobi symbol, and [Formula: see text] is the [Formula: see text]th cyclotomic polynomial in [Formula: see text]. This confirms a recent conjecture of the first author.

We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.

Over the ring of algebraic integers [Formula: see text] of a number field [Formula: see text], the quadratic congruence [Formula: see text] modulo a nonzero ideal [Formula: see text] is considered. We prove explicit formulas for [Formula: see text] and [Formula: see text], the number of incongruent solutions [Formula: see text] and the number of incongruent solutions [Formula: see text] with [Formula: see text] coprime to [Formula: see text], respectively. If [Formula: see text] is contained in a prime ideal [Formula: see text] containing the rational prime [Formula: see text], it is assumed that [Formula: see text] is unramified over [Formula: see text]. Moreover, some interesting identities for exponential sums are proved.

Using a [Formula: see text]-series identity of Carlitz and a [Formula: see text]-analogue of Morley’s congruence due to Pan, we give a proof of a [Formula: see text]-congruence conjectured by Tauraso.

Let [Formula: see text] denote the number of overpartitions of [Formula: see text]. Recently, several infinite families of congruences modulo 12, 27 and 40 for [Formula: see text] have been established. In this paper, we investigate systematically some other infinite families of congruences modulo 12, 24 and 40 for [Formula: see text].

We show that integral solutions of [Formula: see text], where [Formula: see text] is prime, satisfy certain congruence conditions.

We consider two kinds of congruence proposed recently by Sun. The first one is of the form [Formula: see text], where [Formula: see text] is a prime, [Formula: see text] is a character and [Formula: see text] is a sequence of integers. By using a constant term expression for [Formula: see text], we derive congruence for two sequences [Formula: see text]. The second one involves the sum [Formula: see text] where [Formula: see text] By giving a constant term expression for [Formula: see text], we derive a congruence for [Formula: see text] modulo a prime [Formula: see text].

Recently, Lin introduced a new partition function [Formula: see text], which counts the total number of tagged parts over all partitions of [Formula: see text] with designated summands. Lin also proved some congruences modulo [Formula: see text] and [Formula: see text] for [Formula: see text]. In this paper, we shall present two new infinite families of congruences modulo [Formula: see text] for [Formula: see text].

The Delannoy numbers and Schröder numbers are given by [Formula: see text] respectively. We mainly prove that for any prime [Formula: see text], [Formula: see text] which was originally conjectured by Sun in 2011. A related congruence on the Delannoy numbers is also proved.

Suppose that [Formula: see text] is an odd prime and [Formula: see text] is a positive integer with [Formula: see text]. Then for any integer [Formula: see text], [Formula: see text] where [Formula: see text] is the least non-negative remainder of [Formula: see text] modulo [Formula: see text]. This confirms a conjecture of Guo and Zeng.

Let [Formula: see text] be the Franel numbers given by [Formula: see text], and let [Formula: see text] be a prime. In this paper, we mainly determine [Formula: see text] for [Formula: see text].

Wang and Cai established the following harmonic congruence for odd prime and positive integer [Formula: see text]: [Formula: see text] where [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, the authors generalize it into super congruence for odd prime [Formula: see text] and positive integer [Formula: see text]: [Formula: see text] where [Formula: see text] [Formula: see text] [Formula: see text] and [Formula: see text] are positive integers such that [Formula: see text]

Recently, Apagodu and Zeilberger have obtained a congruence involving the Catalan numbers. In this note, we establish a [Formula: see text]-analogue of this congruence.

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

We determine the normalizer in SL2(ℝ) of several families of congruence subgroups of SL2(ℤ). In addition, we show how these tools can be used to evaluate the groups of automorphisms and the discriminant kernels of a large family of isotropic lattices of signature (2,1).

In this paper, we compute constant terms of Eisenstein series defined over a totally real field, at various cusps. In his paper published in 2003, M. Ohta computed the constant terms of Eisenstein series of weight two over the field of rational numbers, at all equivalence classes of cusps. As for Eisenstein series defined over a totally real field, S. Dasgupta, H. Darmon and R. Pollack calculated the constant terms at particular (not all) equivalence classes of cusps in 2011. We compute constant terms of Eisenstein series defined over a general totally real field at all equivalence classes of cusps, and describe them explicitly in terms of Hecke $L$-functions. This investigation is motivated by M. Ohta's work on congruence modules related to Eisenstein series defined over the field of rational numbers., 22 pages

We show that the sum of the traces of Frobenius elements of Artin [Formula: see text]-functions in a family of [Formula: see text]-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for [Formula: see text] and [Formula: see text]-fields. We also prove a central limit theorem for the [Formula: see text]-functions of modular forms on congruence subgroups [Formula: see text] as [Formula: see text].

In recent years, the congruence $$ \sum_{\substack{i+j+k=p\\ i,j,k>0}} \frac1{ijk} \equiv -2 B_{p-3} \pmod{p}, $$ first discovered by the last author have been generalized by either increasing the number of indices and considering the corresponding supercongruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar supercongruences modulo prime powers $p^r$ with the indexes summing up to $mp^r$ where $m$ is coprime to $p$, where all the indexes are also coprime to $p$., 16 pages, final version for publication

Let [Formula: see text] denote the number of overpartitions of [Formula: see text]. In this paper, we will give a complete determination of [Formula: see text] modulo [Formula: see text] by relating it to some binary quadratic forms; this will generalize the result of Kim on [Formula: see text] modulo [Formula: see text]. Moreover, we give infinitely many new Ramanujan-like congruences for overpartition function modulo [Formula: see text].

Let p > 3 be a prime, and let a be a rational p-adic integer. Let {Bn} and {Bn(x)} denote the Bernoulli numbers and Bernoulli polynomials given by B0 = 1,∑k=0n−1n k Bk = 0 (n ≥ 2) and Bn(x) =∑k=0nn k Bkxn−k(n ≥ 0). In this paper, we show that ∑k=0p−1 a k −1 − a k 1 2k − 1 ≡−(2a + 1)(2t + 1) − p2t(t + 1)(4 + (2a + 1)B p−2(−a))(modp3), ∑k=0p−1 a k −1 − a k 2a + 1 2k + 1 ≡ 1 + 2t + p2t(t + 1)B p−2(−a)(modp3), where t = (a −〈a〉p)/p and 〈a〉p ∈{0, 1,…,p − 1} is given by a ≡〈a〉p(modp).

Let [Formula: see text] be a prime. Back in 1878, Lucas proved that the congruence [Formula: see text] holds for any nonnegative integer [Formula: see text]. The converse statement was given in Problem 1494 of Mathematics Magazine proposed in 1997 by Deutsch and Gessel. In this note, we generalize this converse assertion as follows: If [Formula: see text] and [Formula: see text] are integers such that [Formula: see text] for every integer [Formula: see text], then [Formula: see text] is a prime and [Formula: see text] is a power of [Formula: see text].

We give q-analogues of the following congruences by Z.-W. Sun: \sum_{k=1}^{p-1}\frac{D_k}{k} \equiv -\frac{2^{p-1}-1}{p} \pmod p,\\ \sum_{k=1}^{p-1}\frac{H_k}{k 2^k}\equiv 0 \pmod{p},\quad p\geqslant 5, where p is a prime, D_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k} are the Delannoy numbers, and H_n=\sum_{k=1}^n\frac{1}{k} are the harmonic numbers. We also prove that, for any positive integer m and prime p>m+1, \sum_{1\leqslant k_1\leqslant \cdots \leqslant k_m\leqslant p-1}\frac{1}{k_1\cdots k_m 2^{k_m}} \equiv\frac{1}{2}\sum_{k=1}^{p-1}\frac{(-1)^{k-1}}{k^m} \pmod p, which is a multiple generalization of Kohnen's congruence. Furthermore, a q-analogue of this congruence is established., 12 pages

We use bounds of mixed character sum modulo a prime p to study the distribution of points on the hypersurface [Formula: see text] for some polynomials fi ∈ ℤ[X] that are not constant modulo a prime p and integers ki with gcd (ki, p-1) = 1, i = 1, …, n. In the case of [Formula: see text] the above congruence is known as the Markoff–Hurwitz hypersurface, while for [Formula: see text] it is known as the Dwork hypersurface. In particular, we obtain non-trivial results about the number of solution in boxes with the side length below p1/2, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, $A(n,d)$, are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of $A(n,d)$ modulo a prime $\ell$ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial., Comment: 14 pages; preprint of article published in IJNT

In 2007, Andrews and Paule introduced the family of functions Δk(n) which enumerate the number of broken k-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n+1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by 𝔅k(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, [Formula: see text] We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function 𝔅7.

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z.253 (2006) 735–752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler–Zagier correspondences, Trans. Amer. Math. Soc.352 (2000) 2601–2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168–185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math.411 (1990) 66–95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].

We investigate the solubility of the congruence xy ≡ 1 ( mod p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.

Let {En} be the Euler numbers. In the paper we determine E2mk+b – Eb modulo 2m+7, where k and m are positive integers and b ∈ {0, 2, 4, …}.

We study the distribution of solutions n to the congruence σ(n) ≡ a ( mod n). After excluding obvious families of solutions, we show that the number of these n ≤ x is at most x½+o(1), as x → ∞, uniformly for integers a with ∣a∣ ≤ x¼. As a concrete example, the number of composite solutions n ≤ x to the congruence σ(n) ≡ 1 ( mod n) is at most x½+o(1). These results are analogues of theorems established for the Euler ϕ-function by the third-named author.

Let F be the cubic field of discriminant –23 and let [Formula: see text] be its ring of integers. By explicitly computing cohomology of congruence subgroups of [Formula: see text], we computationally investigate modularity of elliptic curves over F.

Let n ≥ 3 be an integer. For any permutation σ of A = {0, 1, …, n-1}, set [Formula: see text]. It is proved that n is prime if and only if 〈σ, n〉 ≡ 0 (mod n) for any σ. When n is composite, we show that there exist cycles of any length satisfying this congruence. We show as well that the analog statement is true for cycles of length at least 2 not satisfying the congruence.

In this paper, we prove arithmetic properties modulo 5 and 7 satisfied by the function pod (n) which denotes the number of partitions of n wherein odd parts must be distinct (and even parts are unrestricted). In particular, we prove the following: for all n ≥ 0, [Formula: see text] and [Formula: see text]

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ( mod p).

In this brief note, we prove one unexpected Ramanujan type congruence modulo 7 for the number cϕ4(n) of generalized Frobenius partitions of n with four colors.

Zhao established the following harmonic congruence for prime p > 3: [Formula: see text] In this note, the authors improve it to the following congruence for prime p > 5: [Formula: see text] Meanwhile, they also improve a generalization of Zhao's congruence and extend Eisenstein's congruence.

Let p be any prime and a be a positive integer. For l, n ∈ {0,1,…} and r ∈ ℤ, the normalized cyclotomic ψ-coefficient [Formula: see text] is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan is often optimal, and two conjectures of Sun and Davis are true.

In this paper, we consider modular forms f(z) whose q-series expansions ∑b(n)qn have coefficients in a localized ring of algebraic integers [Formula: see text]. Extending results of Serre and Ono, we show that if f has non-positive weight, a congruence of the form b(ℓn + a) ≡ 0 (mod ν), where ν is a place over ℓ in [Formula: see text], can hold for only finitely many primes ℓ ≥ 5. To obtain this, we establish an effective bound on ℓ in terms of the weight and the structure of f(z).