On a Diophantine equation with five prime variables

Diophantine Inequalities with Primes, Auxiliary Inequalities, Evaluations of the Difference between Consecutive Primes

The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre - Ingham's problem on primes in intervals $((n - 1)^k, n^k)$ and also a problem on primes in intervals $(\frac{k-1}{k}n, \frac{k}{k-1}n)$ when $k$ is a real number. A number of the new results including an alternative proof of Ingham's theorem with the effectively computable constant and also Ingham's theorem with two primes are proved.

A ternary diophantine inequality by primes near to squares

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.

Some Diophantine equations and inequalities with primes

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.

Diophantine approximation on lines in \mathbb{C}^2 with Gaussian prime constraints - enhanced version

We study the problem of Diophantine approximation on lines in $\mathbb{C}^2$ with numerators and denominators restricted to Gaussian primes. To this end, we develop analogs of well-known results on small fractional parts of $pγ$, $p$ running over the primes and $γ$ being a fixed irrational, for Gaussian primes.

涉及梅森素数的丢番图方程的通解

General Solution of the Diophantine equation involving Mersenne Prime

该研究针对含梅森素数的指数丢番图方程$M_p^{x} + (M_q + 1)^{y}= (lz)^2$展开求解，给出了方程的解的相关示例与无解的情形，丰富了梅森素数相关丢番图方程的研究成果。

On Two Diophantine Inequalities Over Primes

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$ Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3,\,p_4,\,p_5,\,p_6$ for sufficiently large real number $N$.

Diophantine Approximation with Products of Two Primes

We show that for any irrational $α$ and any $τ<8/23$ there are infinitely many $n$ which are the product of two primes for which $$\|nα\|\leq n^{-τ}.$$ We also show that for all sufficiently large $b$ there exist 3-digit palindromes in base $b$ with precisely two prime factors.

Diophantine approximation by special primes

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.

A Diophantine inequality with four prime variables

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.

On a diophantine inequality with prime numbers of a special type

We consider the Diophantine inequality \[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| < (\log N)^{-E} , \] where $1 < c < \frac{15}{14}$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1 + 2, p_2 + 2, p_3 + 2$ has at most $\left[ \frac{369}{180 - 168 c} \right]$ prime factors, counted with the multiplicity.

A quaternary diophantine inequality by prime numbers of a special type

Let $1<c<832/825$. For large real numbers $N>0$ and a small constant $\vartheta>0$, the inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$ such that, for each $i\in\{1,2,3,4\}$, $p_i+2$ has at most $32$ prime factors.

对角丢番图方程组的素数解

Prime Solutions of Diagonal Diophantine Systems

该研究通过建立对角丢番图方程组素数解的渐近公式，结合辛格罗多夫定理和霍利工作，给出了沃林-戈尔德巴奇问题的条件结果。

A Diophantine problem with a prime and three squares of primes

We prove that if $λ_1$, $λ_2$, $λ_3$ and $λ_4$ are non-zero real numbers, not all of the same sign, $λ_1 / λ_2$ is irrational, and $\varpi$ is any real number then, for any $\eps > 0$ the inequality $ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^2 + λ_4 p_4^2 + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-1 / 18 + \eps} $ has infinitely many solution in prime variables $p_1$, ..., $p_4$

两个素数和一个素数k次幂的丢番图逼近问题

A Diophantine approximation problem with two primes and one $k$-th power of a prime

本文改进了两个素数和一个素数k次幂的丢番图逼近问题，将k的范围扩展到1 < k ≤ 3，并给出了更紧的逼近界。

On the existence of $S$-Diophantine quadruples

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.

Almost prime solutions to diophantine systems of high rank

Let $\F$ be a family of $r$ integral forms of degree $k\geq 2$ and $\LL=(l_1,\ldots,l_m)$ be a family of pairwise linearly independent linear forms in $n$ variables $\x=(x_1,...,x_n)$. We study the number of solutions $\x\in[1,N]^n$ to the diophantine system $\F(\x)=\vv$ under the restriction that $l_i(\x)$ has a bounded number of prime factors for each $1\leq i\leq m$. We show that the system $\F$ have the expected number of such "almost prime" solutions under similar conditions as was established for existence of integer solutions by Birch.

Diophantine approximation with one prime of the form $p=x^2+y^2+1$

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 $p_3=x^2 + y^2 +1$.

Diophantine equations in primes: density of prime points on affine hypersurfaces

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…

On two Diophantine inequalities over primes (II)

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 $1<c<\frac{37}{18},c\neq2$.

On a Diophantine problem with one prime, two squares of primes and $s$ powers of two

We refine a result of W.P. Li and Wang on the values of the form $ λ_1p_1 + λ_2p_2^{2} + λ_3p_3^{2} + μ_1 2^{m_1} +...+ μ_s 2^{m_s}, $ where $p_1,p_2,p_3$ are prime numbers, $m_1,..., m_s$ are positive integers, $λ_1,λ_2,λ_{3}$ are nonzero real numbers, not all of the same sign,$λ_2 / λ_3$ is irrational and $λ_i/μ_i \in \Q$, for $i\in\{1,2,3\}$.

Diophantine equations in the primes

Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$ polynomials with integer coefficients of degree $d$ in $n$ variables $\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say $\mathbf{s}$, a general local to global type statement is shown via classical Hardy-Littlewood type methods which provides sufficient conditions for the solubility of $\mathfrak{p}(\mathbf{x})=\mathbf{s}$ under the condition that each of the $x_i$'s is prime.

On a Diophantine inequality involving a prime and an almost-prime

We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$ where $r=3,$ $τ=\frac1{118}$, and $λ_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with $λ_2\neq0$ and $0>\frac{λ_1}{λ_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$

Diophantine approximation by Piatetski-Shapiro primes

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 $p_i=[n_i^c]$, $i=1,\,2,\,3$.

On a Diophantine problem with two primes and s powers of two

We refine a recent result of Parsell on the values of the form $λ_1p_1 + λ_2p_2 + μ_1 2^{m_1} + ...m + μ_s 2^{m_s}, $ where $p_1,p_2$ are prime numbers, $m_1,...c, m_s$ are positive integers, $λ_1 / λ_2$ is negative and irrational and $λ_1 / μ_1$, $λ_2/μ_2 \in \Q$.

A Diophantine problem with prime variables

We study the distribution of the values of the form $λ_1 p_1 + λ_2 p_2 + λ_3 p_3^k$, where $λ_1$, $λ_2$ and $λ_3$ are non-zero real number not all of the same sign, with $λ_1 / λ_2$ irrational, and $p_1$, $p_2$ and $p_3$ are prime numbers. We prove that, when $1 < k < 4 / 3$, these value approximate rather closely any prescribed real number.

由具有有限不同素因子数的有理数进行内在迪奥菲安逼近

Intrinsic Diophantine approximation by rationals of height with a bounded number of distinct prime factors

本文研究了由具有有限不同素因子数的有理数逼近的自相似集的Hausdorff维数，修正了前一版本中引理3.8的证明错误。

素数中的随机丢番图方程 II

Random Diophantine Equations in the Primes II

研究素数中高斯丢番图方程的局部-全局原理，采用Browning等人的方法扩展前人工作。

GRH下二次数域中素数分母的丢番图逼近

Diophantine approximation with prime denominator in quadratic number fields under GRH

本文在广义黎曼猜想对Hecke $L$-函数成立的条件下，将Matomäki关于实数无理数素数分母丢番图逼近的结果推广至所有二次数域。

Piatetski-Shapiro素数的丢番图逼近

Diophantine Approximation with Piatetski-Shapiro Primes

本文证明了关于Piatetski-Shapiro素数的丢番图逼近结果，即在特定参数条件下存在无穷多个形如$p=[n^c]$的素数满足给定的逼近不等式。

Elliptic curves of twin-primes over Gauss field and Diophantine Equations

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_σ: y^2 = x (x+σp)(x+σq) . (σ=\pm 1). E=E_σis also denoted as E_+ or E_- when σ= +1or $-1.Here the Mordell-Weil group and the rank of the elliptic curve E over the Gauss field K=Q(\sqrt -1) (and over the rational field Q is determined in several cases; and results on solutions of related Diophantine equations and simultaneous Pellian equations will be given. The arithmetic constructs over Q of the elliptic curve E have been studied in the last paper1, the Selmer groups are determined, results on Mordell-Weil group,…

素数中的随机丢番图方程

Random Diophantine Equations in the Primes

研究素数解的存在性，提出素数哈塞原理，证明当s≥3k+2时几乎所有此类方程有解，并给出反例显示哈塞-米诺夫斯基定理的素数解版本失效。

Solving diophantine equations x^4 + y^4 = q z^p

We give a method to solve generalized Fermat equations of type $x^4 + y^4 = q z^p$, for some prime values of $q$ and every prime $p$ bigger than 13. We illustrate the method by proving that there are no solutions for $q= 73, 89$ and 113.

用短区间中的素数进行丢番图逼近

Diophantine approximation with primes from short intervals

本文研究了短区间中素数在丢番图逼近中的渐进行为，证明了当素数p满足||pα||<δ时，其数量渐近等于2δY/log X。

On a ternary Diophantine problem with mixed powers of primes

Let $1 < k < 33 / 29$. We prove that if $λ_1$, $λ_2$ and $λ_3$ are non-zero real numbers, not all of the same sign and that $λ_1 / λ_2$ is irrational and $\varpi$ is any real number, then for any $\eps > 0$ the inequality $ \bigl|λ_1 p_1 + λ_2 p_2^2 + λ_3 p_3^k + \varpi \bigr| \le \bigl(\max_j p_j \bigr)^{-(33 - 29 k) / (72 k) + \eps} $ has infinitely many solutions in prime variables $p_1$, ..., $p_k$.

On the solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ for $d$ a prime power

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.

关于具有在$\mathbb{Z}_p^2$-扩张上潜在良好普通减的Abel曲线上Iwasawa理论的结果及其在丢番图稳定性及整数丢番图扩张中的应用

On Iwasawa theory of abelian varieties over $\mathbb{Z}_p^2$-extension with applications to Diophantine stability and integally Diophantine extensions

本文研究Abel曲线上Iwasawa理论，探讨丢番图稳定性及整数丢番图扩张，并改进Mazur猜想相关结果，扩展至具有良好超奇异减的椭圆曲线。

关于素数满足与给定数之和为无平方数的一个丢番图不等式

On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers

本文证明对任意无理数α和实数β，存在无穷多个素数p满足αp+β的分数部分小于p^{-1/10}，且p与给定互异正整数之和均为无平方数。

一个素数、两个素数平方与一个素数$k$次幂的丢番图逼近

Diophantine approximation with one prime, two squares of primes and one $k$-th power of a prime

本研究针对特定范围的$k$与满足条件的非零实系数，证明含一个素数、两个素数平方及一个素数$k$次幂的线性型逼近实数的不等式有无穷多素数解，推进了丢番图逼近领域的相关研究。

关于一个具有形式 $\mathbf{p=x^2+y^2+1}$ 的素数的三元丢番图不等式

A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$

本文证明对任意 $1<c<427/400$，三元 Piatetski-Shapiro 不等式在素数中有解，且其中一个素数可表示为 $x^2+y^2+1$，为此建立了新的 Bombieri--Vinogradov 型指数和结果。

椭圆曲线的算术统计与丢番图稳定性

Arithmetic statistics and diophantine stability for elliptic curves

本文研究有理数域椭圆曲线在素数幂循环扩张中的Mordell-Weil群与Tate-Shafarevich群稳定性，提出“Sha”-稳定性概念，并利用算术统计与岩泽理论证明非CM椭圆曲线在几乎所有素数处稳定。

关于一类涉及特殊类型素数的二元丢番图不等式

On a binary Diophantine inequality involving primes of a special type

本文研究了涉及特殊类型素数的丢番图不等式可解性问题，证明了二元与四元情形下，满足对应素因子个数上界条件的素数解的存在性，给出了素因子数的具体上界公式。

S-Diophantine quadruples with $\mathbf{S=\{2,q\}}$

Let $S$ denote a set of primes and let $a_1,\ldots,a_m$ be positive distinct integers. We call the $m$-tuple $(a_1,\ldots,a_m)$ an $S$-Diophantine tuple if $a_ia_j+1=s_{i,j}$ are $S$-integers for all $i\not=j$. In this paper, we show that no $S$-Diophantine quadruple (i.e~$m=4$) exists if $S=\{2,q\}$ with $q\equiv 3\; (\bmod\, 4)$ or $q<10^9$. For two arbitrary primes $p,q<10^5$ we gain the same result.

混合幂Piatetski-Shapiro素数的丢番图逼近

Diophantine approximation with mixed powers of Piatetski-Shapiro primes

本文研究了混合幂Piatetski-Shapiro素数的丢番图逼近问题，证明在特定条件下存在无限多个素数三元组满足特定不等式。

Diophantine approximation with prime restriction in function fields

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…

超越集对于混合素数幂的丢番图不等式

The exceptional set for Diophantine inequality with mixed powers of primes

研究混合素数幂的丢番图不等式超越集的上界，采用非退化序列和正数δ，确定素数解不存在的上界。

一个素数的五元丢番图不等式，其中一项具有形式 $\mathbf{p=x^2+y^2+1}$

A quinary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$

本文证明了在 $1<c<\frac{5363}{3900}$ 时，存在素数解满足五元丢番图不等式，且其中一素数具有 $p=x^2+y^2+1$ 的形式。

Diophantine problems over $\mathbb{Z}^{ab}$ modulo prime numbers

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.

Inhomogeneous Diophantine approximation in the coprime setting

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 …

素数丢番图逼近与Landau–Siegel零点

Diophantine approximation by primes and Landau--Siegel zeros

本文在假设Siegel零点存在或足够强的条件下，证明了有限型无理数$\alpha$与实数$\beta$的线性组合$\alpha p+\beta$可被素数$p$逼近到$p^{-1/3+\varepsilon}$或$p^{-1/3-1/20}$的精度，给出了素数丢番图逼近的新结果。

Small prime solutions to cubic Diophantine equations

Let a1,..., a9 be non-zero integers and n any integer. Suppose that a1 + ... + a9 = n (mod 2) and (ai, aj) = 1 for 1 <= i < j <= 9. We will prove that (i) if not all of the aj's are of the same sign, then the cubic diagonal equation a1p1^3 + ... + a9p9^3 = n has prime solutions satisfying pj << n^{1/3} + max{|aj|}^{20+ e}; and (ii) if all aj are positive and n >> max{|aj|}^{61+e}, then the cubic diagonal equation a1p1^3 + ... + a9p9^3 = n is soluble in primes pj. This result is motivated from the analogous result for quadratic diagonal equations by S.K.K. Choi and J. Liu. To prove the results …

Gaps between prime divisors and analogues in Diophantine geometry

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.

On the Solutions of the Diophantine Equation $x^n + y^n = z^n$ In the Finite Fields $\mathbb{Z}_p$

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 $\mathbb{Z}_p$.

Formalizing a Diophantine Representation of the Set of Prime Numbers

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…

Note on the diophantine equation X^t+Y^t=BZ^t

In this paper, we obtain new results on the integers solutions X, Y, Z of the diophantine equation X^t+Y^t=BZ^t for a rationnal integer B and a prime number t verifying some conditions explained in the paper.