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According to Skolem's conjecture, if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriately chosen modulus. Besides several concrete equations, the conjecture has only been proved for rather…

We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a…

In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

We prove a theorem about approximation to an irrational number by rational numbers whose denominator n is free of prime factors bigger than a power of log n. We strengthen the result in version 1 by using an exponential sum over smooth…

We show the insolvability of the Diophantine equation $ax^d-y^2-z^2+xyz-b=0$ in $\mathbb{Z}$ for fixed $a$ and $b$ such that $a\equiv 1 \pmod {12}$ and $b=2^da-3$, where $d$ is an odd integer and is a multiple of $3$. Further, we…

We investigate unique continuation properties and asymptotic behaviour at boundary points for solutions to a class of elliptic equations involving the spectral fractional Laplacian. An extension procedure leads us to study a degenerate or…

The study of finiteness or infiniteness of integer solutions of a Diophantine equation has been considered as a standard problem in the literature. In this paper, for f(x) in Z[x] monic and q1 ,...., qm in Z, we study the conditions for…

Let $(a,b,c)$ be a primitive Pythagorean triple. Set $a=m^2-n^2$,$b=2mn$, and $c=m^2+n^2$ with $m$ and $n$ positive coprime integers, $m>n $ and $ m \not \equiv n \pmod 2$. A famous conjecture of Je\'{s}manowicz asserts that the only…

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most…

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)^\sigma u = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We…

For fixed integers $D \geq 0$ and $c \geq 3$, we demonstrate how to use $2$-adic valuation trees of sequences to analyze Diophantine equations of the form $x^2+D=2^cy$ and $x^3+D=2^cy$, for $y$ odd. Further, we show for what values $D \in…

The title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there are…

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(\lambda)+1/y(\lambda)+1/z(\lambda)$ with $n=b+a\lambda$ are explicitly given for a,b coprime and a not a multiple of m . The…

By defining $$I_n:=\int_{0}^{1}\int_{0}^{1} \frac{(x(1-x)y(1-y))^n}{(1-xy)(-\log xy)}\ dx dy$$ Sondow (see [2]) proved that $$I_n=\binom{2n}{n} \gamma+L_n-A_n$$ We prove asymptotic formula for $L_n$ and $A_n$ as $n\to\infty$, $$…

This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, \begin{equation*} \mbox{div} (a_+ \nabla u^+ - a_- \nabla u^-) = \mbox{div} f, \end{equation*} where $a_+$ and $a_-$ are…

We review the indefinite sublinear elliptic equation $-\Delta u=a(x)u^{q}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}$, with Dirichlet or Neumann homogeneous boundary conditions. Here $0<q<1$ and $a$ is continuous and changes…

We analyze an elliptic equation arising in the study of the gauged O(3) sigma model with the Chern-Simons term. In this paper, we study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However,…

By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and…

Given integers $a,b>1$ with $\log_b{a}$ irrational, we investigate the connection between the conjectured asymptotic equidistribution of digits in the base-$b$ expansion of $a^n$ and the (non-)Diophantine properties of the number…