In this paper we study the Diophantine equation $x^{4}-q^{4}=py^{5},$ with the following conditions: $p$ and $q$ are different prime natural numbers, $y$ is not divisible with $p$, $p\equiv3$ (mod20), $q\equiv4$ (mod5), $\overline{p}$ is a generator of the group $(U(\textbf{Z}_{q^{4}}),\cdot)$, $(x,y)=1$, 2 is a 5-power residue mod $q$.
Diophantine Inequalities as a Problem of Difference between Consecutive Primes
在本文中，我们已经开发了一种方法来解决\textit{diophantine inequalities}使用它们与\textit{difference between consecutive primes}的关系。使用这种方法，我们已经能够证明一些定理，包括英厄姆的指数定理以及一些新的结果。丢番图不等式及其与Cramer和Andrica不等式的关系也被讨论。
Given a finite set of primes $S$ and a $m$-tuple $(a_1,\dots,a_m)$ of positive, distinct integers we call the $m$-tuple $S$-Diophantine, if for each $1\leq i < j\leq m$ the quantity $a_ia_j+1$ has prime divisors coming only from the set $S$. For a given set $S$ we give a practical algorithm to find all $S$-Diophantine quadruples, provided that $|S|=3$.
Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,\,p_2,\,p_3$ near to squares.
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 solutions in primes exist.
Let $[x]$ denote the integral part of the real number $x$, and $N$ be a sufficiently large integer. In this paper, it is proved that, for $1<c<\frac{4109054}{1999527}, c\not=2$, the Diophantine equation $N=[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]$ is solvable in prime variables $p_1,p_2,p_3,p_4,p_5$.
We show that whenever $δ>0$, $η$ is real and constants $λ_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|λ_1p_1 + λ_2p_2 + λ_3p_3+η|<(\max p_j)^{-1/12+δ}$ and such that, for each $i\in\{1,2,3\}$, $p_i+2$ has at most $28$ prime factors.
Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4$, which improves the result of Mu.
Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers $s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mod 4$ no $\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a Wieferich prime pair.
Let $\varepsilon>0$ be a small constant. In the present paper we prove that whenever $η$ is real and constants $λ_i$ satisfy some necessary conditions, then there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<\varepsilon \end{equation*} and such that $p_3=x^2 + y^2 +1$.
Let $F \in \mathbb{Z}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d \geq 2$, and let $V_F^*$ denote the singular locus of the affine variety $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. In this paper, we prove the existence of integer solutions with prime coordinates to the equation $F(x_1, \ldots, x_n) = 0$ provided $F$ satisfies suitable local conditions and $n - \dim V_F^* \geq 2^8 3^4 5^2 d^3 (2d-1)^2 4^{d}$. Our result improves on what was known previously due to Cook and Magyar (B. Cook and A. Magyar, `Diophantine equations in the primes'. Invent. Math. 198 (2…
Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N \end{equation*} is solvable in primes $p_1,p_2,p_3$. Moreover, we also prove that the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5,p_6$, which improves the previous result $1<c<\frac{37}{18},c\neq2$.
Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $η$ is real, the constants $λ_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |λ_1p_1 + λ_2p_2 + λ_3p_3+η|<(\max p_j)^{{\frac{37c-38}{26c}}}(\log\max p_j)^{10} \end{equation*} and such that $p_i=[n_i^c]$, $i=1,\,2,\,3$.
In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the characterization of primitive divisors on Lehmer sequences and the development of an algorithmic method of proving the non-existence of integer solutions of the equation $f(x)=a^b$, where $f(x)\in\mathbb Z[x]$, $a$ a positive integer and $b$ an arbitrary positive integer.
In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $pα$ when $α$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to the nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered t…
Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients, has a solution in $\mathbb{Z}^{ab}$ modulo every rational prime.
Given $n\in N$ and $x,γ\in R$, let \begin{equation*} ||γ-nx||^\prime=\min\{|γ-nx+m|:m\in Z, \gcd (n,m)=1\}, \end{equation*} %where $(n,m)$ is the largest common divisor of $n$ and $m$. Two conjectures in the coprime inhomogeneous Diophantine approximation state that for any irrational number $α$ and almost every $γ\in R$, \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime}=0 \end{equation*} and that there exists $C>0$, such that for all $α\in R\backslash Q$ and $γ\in [0,1)$ , \begin{equation*} \liminf_{n\to \infty}n||γ-nα||^{\prime} < C. \end{equation*} We prove the first conjecture and …
Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.
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 in this paper is to give optimal conditions or relationships between the exponent $n$ and the prime $p$ to determine the existence of nontrivial solutions of the diophantine equation $x^n + y^n = z^n$ with $1 \leq n \leq p -1 $, in finite fields $\mathbb{Z}_p$.
The DPRM (Davis-Putnam-Robinson-Matiyasevich) theorem is the main step in the negative resolution of Hilbert's 10th problem. Almost three decades of work on the problem have resulted in several equally surprising results. These include the existence of diophantine equations with a reduced number of variables, as well as the explicit construction of polynomials that represent specific sets, in particular the set of primes. In this work, we formalize these constructions in the Mizar system. We focus on the set of prime numbers and its explicit representation using 10 variables. It is the smalles…
