Related papers: A Polynomial Diophantine Generator Function for In…

The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal…

We describe a bound on the degree of the generators for some adjoint rings on surfaces and threefolds.

We compute the numbers g(n,2,2) of nilpotent groups of order n, of class at most 2 generated by at most 2 generators, by giving an explicit formula for the Dirichlet generating function \sum_{n=1}^\infty g(n,2,2)n^{-s}.

We investigate a family of Diophantine polynomial equations which involve continuant functions. In particular, given a polynomial $P(x)\in \mathbb{Z}[x]$ and $n\in \mathbb{N}$, we consider the equation $P(K_n(x_1,\ldots, x_n)) =…

Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…

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…

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.

Generalizing an argument of Matiyasevich, we illustrate a method to generate infinitely many diophantine equations whose solutions can be completely described by linear recurrences. In particular, we provide an integer-coefficient…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. For a positive integer n, let f(n) denote the greatest finite total number of solutions of a subsystem of E_n in integers x_1,...,x_n. We prove: (1) the function f is…

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions…

A classical method for partition generating functions is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n copies of n are presented.

We present an approach for construction of functional bases of differential invariants for some infinite-dimensional algebras with coefficients of generating operators depending on arbitrary functions. An example for the…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. There is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer…

Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer solutions and check for…

Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper we study Diophantine graphs, i.e., finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only…

We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…

We use the Circle Method to derive asymptotic formulas for functions related to the number of parts of partitions in particular residue classes.

For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ of smallest degree $n$ that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their…

The paper assesses the top number of integer solutions for algebraic Diophantine Thue diagonal equation of the degree $n \geq 2$ and number of variables $k > 2$ and equations with explicit variable in the case when the coefficients of the…

In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements. Our method allows us not only to…