Related papers: On an tangent equation by primes

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.

The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many…

Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this…

We show that whenever $\delta>0$, $\eta$ is real and constants $\lambda_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|\lambda_1p_1 + \lambda_2p_2 +…

Let $k\geq2$ and $s$ be positive integers. Let $\theta\in(0,1)$ be a real number. In this paper, we establish that if $s>k(k+1)$ and $\theta>0.55$, then every sufficiently large natural number $n$, subjects to certain congruence conditions,…

Let $d$ be a positive integer. Let $p$ be a prime number. Let $\alpha$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $\xi$ of degree $d$ such that…

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\,n_2,\,n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split…

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1, n_2, n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split…

This paper studies integer solutions to the Diophantine equation A+B=C in which none of A, B, C have a large prime factor. We set H(A, B,C) = max(|A|, |B|, |C|), and consider primitive solutions (gcd}(A, B, C)=1) having no prime factor p…

Let $1<k<14/5$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality…

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…

Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, combining the Gel'fond-Baker method with an elementary approach, we prove that if $\max\{a,b,c\}>5\times 10^{27}$, then the equation $a^x+b^y=c^z$ has at…

Let $c$ be a square-free positive integer and $p$ a prime satisfying $p\nmid c$. Let $h(-c)$ denote the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-c})$. In this paper, we consider the Diophantine equation…

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…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denote 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 $\frac{13}{14}<\gamma<1$, there…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

We deeply investigate the Diophantine equation $cx^2+d^{2m+1}=2y^n$ in integers $x, y\geq 1, m\geq 0$ and $n\geq 3$, where $c$ and $d$ are given coprime positive integers such that $cd\not\equiv 3 \pmod 4$. We first solve this equation for…