Related papers: An Interesting Number Theoretic Problem

A complete p-adic Khintchine type theorem for approximation by p-adic algebraic numbers is established.

The purpose of this paper is to prove the existence of solutions of quasi-equilibrium problems without any generalized monotonicity assumption. Additionally, we give an application to quasi-optimization problems.

We solve multiple conjectures by Byszewski and Ulas about the sum of base $b$ digits function. In order to do this, we develop general results about summations over the sum of digits function. As a corollary, we describe an unexpected new…

For $H \ge N^{1-\frac{1}{2c}} \ln^2 N$, where $c$ is a fixed non-integer number satisfying $$ \|c\| \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N}, \qquad c > \frac{4}{3}\left(1 + \frac{52\ln \ln N}{\ln N}\right), $$ we obtain an…

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements,…

For any three element set of positive integers, $\{a,b,n\}$, with $a<b<n$, $n$ sufficiently large and $\gcd(a,b)=1$, we find the least $\alpha$ such that given any real numbers $t_1$, $t_2$, $t_3$, there is a real number $x$ such that…

For any integer $r \geq 1$, the sequence of numbers $\{{c^{(r)}_{k}}\}_{k \geq 0} $ is defined implicitly by [\sum_k\binom{n}{k}^r\binom{n+k}{k}^r = \sum_k\binom{n}{k}\binom{n+k}{k}c^{(r)}_k,\quad n=0,1,2,...] Asmus Schmidt conjectured that…

We prove that for all integers $k \geq 1$, there exists a constant $C_k$ depending only on $k$, such that for all $q > C_k$, and for $n = 1, 2$ every matrix in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers and for all $n \geq 3$ every…

We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be…

Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.

We consider dynamical systems $(X,T,\mu)$ which have exponential decay of correlations for either H\"older continuous functions or functions of bounded variation. Given a sequence of balls $(B_n)_{n=1}^\infty$, we give sufficient conditions…

Let ${\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\rm…

We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the…

In this note, we continue an approach pursued in an earlier paper of the second author and thereby attempt to produce an improved lower bound for the sum $I(q^k) + I(n^2)$, where $q^k n^2$ is an odd perfect number with special prime $q$ and…

A systematic study of the trigonometric equation A tan a + B sin b = C, where A, B and C^2 are rational numbers. The special case tan Pi/11 + 4 sin 3 Pi/11 = sqrt[11] appears in the classical literature.

This note presents a unified theorem of the alternative that explicitly allows for any combination of equality, componentwise inequality, weak dominance, strict dominance, and nonnegativity relations. The theorem nests 60 special cases,…

In this paper, we investigate some congruences involving sums of $\frac{d^{-k}{x\choose k}{x+k\choose k}}{{2k \choose k}}$, where $x$ be a $p$-adic integer, $k$ be a non-negative integer, and $d$ $(d\neq 0)$ be a rational number.

We study $B(n;k)$, the number of ways of writing $n$ as a sum or difference of the first $k$ Fibonacci numbers. We show that $B(0;k)$ satisfies the Tribonacci-like recurrence $B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2)$ and that $B(n;k)$ satisfies a…

Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero constant term. We show that for any finite coloring of $\mathbb{Q}$, there are non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a set of…