Related papers: Proof of Andrews' conjecture on a_4\phi_3 summatio…

In the recent article arXiv:1606.03351, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end they…

In 2005, the second author and Todorov introduced an upper bound on the finitistic dimension of an Artin algebra, now known as the {\phi}-dimension. The {\phi}-dimension conjecture states that this upper bound is always finite, a fact that…

We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can…

A family of congruences interpolating between those of Wilson and Giuga is constructed. Several elementary results are established, in order to present a possible approach to establishing Giuga's conjecture.

In this paper we show the equivalence of the conjectures of Giuga and Agoh in a direct way which leads to a combined conjecture. This conjecture is described by a sum of fractions from which all conditions can be derived easily.

In this paper, we prove two congruences on the double sums of the super Catalan numbers (named by Gessel), which were recently conjectured by Apagodu.

For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a…

We propose a conjectural formula for $DR_g(a,-a) \lambda_g$ and check all its expected properties. Our formula refines the one point case of a similar conjecture made by the first named author in collaboration with Gu\'er\'e and Rossi, and…

We give a new proof of Brooks' theorem that immediately implies a strengthening of Brooks' theorem, known as Catlin's theorem.

We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical…

Let A be an abelian fourfold. We prove the Standard Conjecture of Hodge type for A. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with l-adic homological equivalence on A for…

The paper contains an alternative proof of M. Kontsevich Formality Theorem.

In this article we prove that equation $\phi(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product…

We prove a formal power series identity, relating the arithmetic sum-of-divisors function to commuting triples of permutations. This establishes a conjecture of Franklin T. Adams-Watters.

Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…

Two new generalized Fibonacci number summation identities are stated and proved, and two other new generalized Fibonacci number summation identities are derived from these, of which two special cases are already known in literature.

We discuss several existing proofs of the value of a quartic integral and present a new proof that evolved from rational Landen transformations.

We compute the renormalized trajectory of $\phi^4_4$-theory by perturbation theory in a running coupling. We introduce an iterative scheme without reference to a bare action. The expansion is proved to be finite to every order of…

A proof of Sendov's conjecture is given.

In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Ap\'ery-like numbers modulo $p^3$, where $p$ is an odd prime.