Related papers: A Diophantine inequality with four prime variables

The sufficient conditions for solvability of a linear Diophantine equation $\sum_{i=1}^{n}a_ix_i=b$ (with $a_1,a_2,...,a_n\in \mathbb{N}$) in non-negative integers $x_1,x_2,...,x_n$ are given. The explicit formulas are given for Frobenius…

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation…

In this paper we first show that, under certain conditions, the solution of a single quadratic diophantine equation in four variables $Q(x_1,\,x_2,\,x_3,\,x_4)=0$ can be expressed in terms of bilinear forms in four parameters. We use this…

Matom\"aki proved that if $\alpha\in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\alpha-a/p|\le p^{-4/3+\varepsilon}$ for a suitable integer a. In this paper, we extend this result to all quadratic…

Let $f(x_1,\ldots,x_n)$ be a regular indefinite integral quadratic form with $n\ge 9$, and let $t$ be an integer. It is established that $f(x_1,\ldots,x_n)=t$ has solutions in prime variables if there are no local obstructions.

We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in…

We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…

We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.

This paper makes the following conjecture: For every prime $p$ there exists a positive integer $x$ with $\left\lceil \frac{p}{4} \right\rceil \leq x \leq \left\lceil \frac{p}{2} \right\rceil$ and a positive divisor $d|x^2$ so that either:…

Let $(U_n)_{n\in \mathbb{N}}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants $B$ and $N_0$ such that for any $b,c\in \mathbb{Z}$…

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one…

The main aim of this article is to find all solutions of the Diophantine equation $x^2 + p^k=y^n$ where $p \equiv 1 \pmod 4$, $\frac{p-1}{3}$ is a perfect square and the class number of $\mathbb{Z}[\sqrt{-p}]$ is $2$. In this article, I…

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $\sigma_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for…

We give a necessary condition for the existence of solutions of the Diophantine equation $p=x^{q}+ry^{q},$ with $p$, $q$, $r$ distinct odd prime natural numbers.

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12/11$ there exist infinitely…

An asymptotic formula for the number of prime solutions of a general diagonal system of Diophantine equations is established, contingent on the existence of an appropriate mean value bound and on local solvability. In conjunction with the…

Let $p$ be a prime integer, $\mathbb{Z}_p$ the finite field of order $p$ and $\mathbb{Z}^{*}_{p}$ is its multiplicative cyclic group. We consider the Diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq \frac{p - 1}{2}$. Our main aim…

Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $\alpha$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(\alpha)/\mathbb Q$ is Galois. We prove that, for every…

We show that for any irrational $\alpha$ and any $\tau<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|n\alpha\|\leq n^{-\tau}.$$ We also show that for all sufficiently large $b$ there exist 3-digit…

The subject matter of this work is the diophantine equation x^n+y^m=c(x^k)(y^l), where n,m,k,l,c are natural numbers.We investigate this equation from the point of view of positive integer solutions.A preliminary examination of sources such…