一个加性混合幂丢番图不等式Ⅲ

证明了:如果λ1,…,λ11,μ是非零实数,并且不同一符号,至少有一个λi/λj是无理数,那么对任意实数η和ε>0,不等式λ1x14 … λ11x141 μy2 η<ε有无穷多正整数解x1,…,x11,y.     

Cubic Diophantine Inequalities

Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ _i ≥ 1. In this paper, we prove there are integers x ₁,..., x ₇ such that the inequalities |λ₁ x ³ ₁ + λ₂ x ³ ₂ + ⋯ + λ₇ x ³ ₇| < 1 and hold simultaneously. Received November 18, 1997, Accepted October 23, 1998     

一个混合幂为2和4的丢番图不等式

In this paper,it is shown that:if λ1,..,λ8 are nonzero real numbers,not all of the same sign,such that λ1/λ2 is irrational,then for any real numbex,η and ε＞0 the inequality,|λ1x21 λ2x22 λ3x43 … λ8x48 η|＜ε has infinitely many solutions in positive integers x1,…,x8.     

一个加性混合幂丢番图不等式(Ⅰ)

本文证明了如果λ1,…,λ6是非零实数,并且不同一符号,至少有一个λi/λj(1≤i,j≤3)是无理数,那么对任意实数η和ε＞0,不等式|λ1x21+λ2x22+λ3x23+λ4x44+λ5x45+λ6x46+η|＜ε有无穷多正整数解x1,…,x6.     

关于两个素数和一个素数k次幂的丢番图不等式

令■设λ_1,λ_2,λ_3是不全同号的非零实数,且满足λ_1/λ_2为无理数,则对于任意实数η和ε 0,不等式■有无穷多组素数解p_1,p_2,p_3.该结果改进了Gambini,Languasco和Zaccagnini的结果.     

On the Diophantine Inequality Problem

In this paper,we deal with a Diophantine inequality involving a prime,two squares of primes and one k-th power of a prime which give an improvement of the resul...     

一个二元丢番图不等式

本文证明了如果1相似文献   

The Solubility of Diagonal Cubic Diophantine Equations

This paper proves conditional existence results for non-trivialsolutions of the equation where the coefficients a_i and the unknowns X_i are taken to berational integers. No such results were previously known for n6. The proofs useelementary facts about the 3-descent procedure for ellipticcurves of the form E_A: X³ + Y³ = AZ³. Thus, when n=4, and the a_i are each prime, and are all congruentto 2 modulo 3, it is shown that (*) will have non-trivial solutions,providing that the Selmer conjecture holds for the curves E_A.One may replace the Selmer conjecture by an appropriate formof the Generalized Riemann Hypothesis, when n=5 and the a_i areagain taken to be primes, all congruent to 8 modulo 9. Finally,when n=5, one may require only that the a_i be square-free andcoprime to 3, providing one assumes both the Selmer conjectureand a special case of Schinzel's conjecture (on the representationof primes by cubic polynomials). 1991 Mathematics Subject Classification:11D25, 11G05, 14G05.     

一个素变数丢番图不等式

本文证明了如果１＜ｃ＜２３７／２１４，则对于充分大的Ｎ和ε≥Ｎ－１／５（２３７／２１４－ｃ）＋ｖ（ｖ＞０）表达式Ｄ（Ｎ）：＝Σ｜ｐｃ１＋ｐｃ２＋ｐｃ３－Ｎ｜＜－εｌｏｇｐ１ｌｏｇｐ２ｌｏｇｐ３ 关于素变数ｐ１，ｐ２，ｐ３有渐近公式，改进了文献［１］的结果１＜Ｃ＜１．１．     

一个素变数丢番图不等式

本文证明了如果1相似文献   

On a Diophantine Inequality with Reciprocals

We generalize Chebotarev’s density theorem to Weil groups. Since the Artin–Weil conjecture on the integrality of the Artin–Hecke L-functions, constructed by A. Weil, has not been completely proved so far, we estimate the character sums both under and without the assumption of the validity of that conjecture.     

有关殆素数的二元丢番图不等式

Pr表示最多r个素因子的正整数.作者证明了,对于任一足够大的实数N和1＜c＜c0,不等式|pc+Pcr-N|＜N9/10(1-c0/c)对素数p和Pr是可解的.特别地,当1＜c＜c0=1.03074432…,有r=6.这个结果改进了翟文广和曹晓东的结果.     

一个素变数的Diophantine不等式

设１＜Ｃ＜１３／１２．本文证明了存在Ｎ（Ｃ）＞０使得对任意实数Ｎ＞Ｎ（Ｃ），下面的不等式｜＋＋　－Ｎ｜＜Ｎ－　　　　）ｌｏｇｓＮ有素数解ｐ１，ｐ２，ｐ３，其中ｓ＝２（１５）．     

一个素变数的Diophantine不等式(Ⅲ)

令.本文证明了存在N（c）＞0,对任意实数N＞N（c）,不等式有素数解p1,p2,p3,其中c1为一绝对正常数．     

一个素变数的Diophantine不等式(III)

令     

Cubic Forms Modulo One

By using a more efficient way of ‘almost diagonalizing’,known results on small fractional parts of general real cubicforms in a small number of variables, are improved. In particular,for ternary forms the method gives a bound superior to thatarising from simply reducing to a cubic form in one variable.2000 Mathematics Subject Classification 11D75, 11E76, 11J54.     

On a Binary Diophantine Inequality Involving Prime Numbers

Let 1 < c < 15/14 and N a sufficiently large real number. In this paper we prove that, for all (N, 2N\ A with , the inequality has solutions in primes .     

Two Diophantine Birds with One Stone

Integer Solutions are found to the equations t²–3(a²,b², (a + b)², (a–b)²) = p², q², r², s². These lead surprisinglyto solutions to the equations u² + (c², d², (c + d)², (c –d)²) = p², q², v², w², with the same values of p and q.     

On a Diophantine Inequality Over Primes (II)

In this short note we prove that if 1 < c < 81/40, c ≠ 2, N is a large real number, then the Diophantine inequality is solvable, where p ₁,···,p ₅ are primes.     

一个素变量混合幂丢番图不等式

证明了:设k是大于或等于2的正整数,η是任意给定的实数,λ_1,λ_2,λ_3是非零实数,不全同号,并且λ_1/λ_2是无理数,则不等式|λ_1p_1+λ_2p_2+λ_3p_32~k+η|(max p_j)~(-σ)有无穷多组素数解p_1,p_2,p_3,这里σ满足:当2≤k≤3时,0σ1/2(2~(k+1)+1),当4≤k≤5时,0σ5/6k2~k;当k≥6时,0σ20/21k2~k.