Related papers: Two Triple binomial sum supercongruences

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will…

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…

Using the methodology of (rigorous) {\it experimental mathematics}, we give a simple and motivated solution to Zudilin's question concerning a $q$-analog of a problem posed by Asmus Schmidt about a certain binomial coefficients sum. Our…

We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the…

Recently, Drema and Saikia (2023) proved several congruences modulo powers of 2 and 3 for overpartition triples with odd parts. We extend their list substantially. We prove several congruences modulo powers of 2 for overpartition k-tuples…

In this short note, we prove a conjecture recently posed by Alekseyev, Amdeberhan, Shallit, and Vukusic on the 3-adic valuation of a cubic binomial sum.

We address three questions posed by Bibak \cite{KB20}, and generalize some results of Bibak, Lehmer and K G Ramanathan on solutions of linear congruences $\sum_{i=1}^k a_i x_i \equiv b \Mod{n}$. In particular, we obtain explicit expressions…

In 1997, van Hamme conjectured 13 Ramanujan-type supercongruences labeled (A.2)--(M.2). Using some combinatorial identities discovered by Sigma, we extend (A.2) and (H.2) to supercongruences modulo $p^4$ for primes $p\equiv 3\pmod{4}$,…

Recently, Z.-W. Sun introduced two kinds of polynomials related to the Delannoy numbers, and proved some supercongruences on sums involving those polynomials. We deduce new summation formulas for squares of those polynomials and use them to…

We derive various weighted summation identities, including binomial and double binomial identities, for Tribonacci numbers. Our results contain some previously known results as special cases.

In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…

With the help of El Bachraoui's lemma, the creative microscoping method, and a new form of the Chinese remainder theorem for coprime polynomials, we prove a $q$-supercongruence for double series and a $q$-supercongruence for triple series…

Recently, Nadji, Ahmia and Ram\'{i}rez \cite{Nadji2025} investigate the arithmetic properties of ${\bar B}_{\ell_1,\ell_2}(n)$, the number of overpartitions where no part is divisible by $\ell_1$ or $\ell_2$ with $\gcd(\ell_1,\ell_2)$$=1$…

In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.

We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.

In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et al.} (2002, 2007).

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let $p>3$ be a prime. We prove that…

Let $p$ be a large prime number and $g$ be any integer of multiplicative order $T$ modulo $p$. We obtain a new estimate of the double exponential sum $$ S=\sum_{n\in \mathcal{N}}\left|\sum_{m\in \mathcal{M} }e_p(an g^{m})\right|, \quad \gcd…

We confirm several conjectures of Guo, Jouhet and Zeng concerning the factors of alternative binomials sums.

We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results…