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In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.
The aim of this paper is to investigate different types of multi-integrals of finite variation and to obtain decomposition results.
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein together with theorems corollaries, formulae, examples, mathematical criteria, etc. (about integer sequences, numbers, quotients, residues,…
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.
In this paper we prove an identity in terms of generating functions which enables us to calculate the numbers of isomorphism classes of absolutely indecomposable semistable representations of quivers over finite fields.
Consider $k\ge 2$ distinct, linearly independent, homogeneous linear recurrences of order $k$ satisfying the same recurrence relation. We prove that the recurrences are related to a decomposable form of degree $k$, and there is a very broad…
We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…
In this paper we study the existence of solutions to an isotropic differential inclusion.
An interplay between the sum of certain series related to Harmonic numbers and certain finite trigonometric sums is investigated. This allows us to express the sum of these series in terms of the considered trigonometric sums, and permits…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
We enumerate the 15768 perfect groups of order up to $2\cdot 10^6$, up to isomorphism, thus also completing the missing cases in the prior classification. The work supplements the by now well-understood computer classifications of solvable…
A $\lambda$-quiddity of size $n$ is an $n$-tuple of elements from a fixed set, which is a solution to a matrix equation that arises in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion…
Two types of finite series of products of harmonic numbers involving nonnegative integer powers are evaluated, also yielding two other important harmonic number identities. The recursion formulas for these sums are derived, which are easily…
We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…
Let $n$ be a non-negative integer and put $p_{n}(x)=\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\sum_{i=1}^{k}p_{a_{i}}(x), $$…
In this article we further develop methods for representing integers as a sum of three cubes. In particular, a barrier to solving the case $k=3$, which was outlined in a previous paper of the second author, is overcome. A very recent…