Related papers: Norm Form Equations and Linear Divisibility Sequen…

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive discriminant. In this paper, we consider the Diophantine equation $u_m + u_n = a_1 n_1! + \cdots + a_k n_k!$ and prove that there are only finitely many…

Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study…

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…

A meromorphic solution of a complex linear differential equation (with meromorphic coefficients) for which the value zero is the only possible finite deficient/deviated value is called a standard solution. Conditions for the existence and…

Binary quadratic Diophantine equations are of interest from the viewpoint of computational complexity theory. They contain as special cases many examples of natural problems apparantly occupying intermediate stages in the P-NP hierarchy,…

We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.

A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are…

In this work, the power series solutions are given around a regular-singular point, in the case of variable coefficients for homogeneous sequential linear conformable fractional differential equations of order 2{\alpha}.

The notion of type of a differential 2-form in four variables is introduced and for 2-forms of type < 4, local normal models are given. If the type of a 2-form $\Omega$ is 4, then the equivalence under diffeomorphisms of $\Omega$ is reduced…

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence…

Here we show a hidden regularity result for nonlinear wave equations with an integral term of convolution type and Dirichlet boundary conditions. Under general assumptions on the nonlinear term and on the integral kernel we are able to…

System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…

We derive some regularity estimates of the solution to a time fractional diffusion equation, that are useful for numerical analysis, and partially unravel the singularity structure of the solution with respect to the time variable.

A scalar integer partition problem asks for a number of nonnegative integer solutions to a linear Diophantine equation with integer positive coefficients. The manuscript discusses an algorithm of derivation of linear relations involving the…

We analyze solvability of a special form of distributed order fractional differential equations within the space of tempered distributions supported by the positive half-line.

We give an alternative method to obtain normal forms of reversible equivariant vector fields. We adapt the classical method using tools from invariant theory to establish formulae that take symmetries into account as a starting point.…

We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of…

Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions…

In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…