One Additive Diophantine Inequality with Mixed Powers 2 and 4

In this paper, it is shown that： if λ1 ,……λs axe nonzero real numbers, not all of the same sign, such that A1/A2 is irrational, then for any real number η and ε 〉 0 the inequality ｜λ1x1^2 ＋ λ2x2^2＋ λ3x3^4＋ λsx3^4＋……λsx8^4 ＋η〈 ε has infinitely many solutions in positive integers x1,... ,xs.

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 result given by Alessandro Gambini.

Diophantine inequality involving binary forms

Let r =2^d-1 ＋ 1. We investigate the diophantine inequality｜∑i=1^r λiФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-δ,where Фi（x,y）∈X[x,y]（1≤i≤r） are nondegenerate forms of degree d = 3 or 4.

Diophantine inequality involving binary forms

Let d ≥ 3 be an integer, and set r = 2^d-1 ＋ 1 for 3 ≤ d ≤ 4, r = 17 5~ ＂2441 for 5 ≤ d ≤ 6, r = d^2＋d＋1 for 7 ≤ d ≤ 8, and r = d^2＋d＋2 for d ≥ 9, respectively. Suppose that Фi（x, y） E Z[x, y] （1 ≤ i ≤ r） are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ1, λ2,. ..., λr are nonzero real numbers with λ1/λ2 irrational, and λ1λ1（x1, y1） ＋ λ2q）2（x2, y2） ＋ ... ＋ ）,λrФr（xr, yr） is indefinite. Then for any given real η and σ with 0 〈 cr 〈 22-d, it is proved that the inequalityhas infinitely ｜r∑i=1λФi（xi,yi）＋η｜〈（max 1≤i≤r{｜xi｜,｜yi｜}）^-σmany solutions in integers Xl, x2,..., xr, Yl, Y2,.--, Yr. This result constitutes an improvement upon that of B. Q. Xue.

Diophantine Inequality by Unlike Powers of Primes

Suppose that λ_(1),…,λ_(5) are nonzero real numbers,not all of the same sign,satisfying that λ_(1)/λ_(2) is irrational.Then for any given real number η and ε>0,the inequality |λ_(1)p_(1)+λ_(2)p_(2)^(2)+λ_(3)p_(3)^(3)+λ_(4)p_(4)^(4)+λ_(5)p_(5)^(5)+η|<(max_(1≤j≤5)p_(j)^(j))^(-19/756+ε) has infinitely many solutions in prime variables p_(1),…,p_(5).This result constitutes an improvement of the recent results.

Diophantine Inequalities with Mixed Powers

It is proved that if λ1,λ2,…,λ7 are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 〈 σ 〈 1/16, the inequality ｜λ1x1^2＋λ2x2^2＋∑i=3 7λixi^4＋η｜〈（max1≤i≤7｜xi｜-σhas infinitely many solutions in positive integers xl, x2,... , x7. Similax result is proved for ｜λ1x1^2＋λ2x2^2＋λ3x3^2＋λ4x4^4＋λ5x5^4＋λ6x6^4＋η｜〈（max1≤i≤7｜xi｜-σ.These results constitute an improvement upon those of Shi and Li.

Diophantine Approximation with Four Squares of Primes and Powers of Two

Under certain condition, the inequality |λ_1p_1~2+λ_2p_2~2+λ_3p_3~2+λ_4p_4~2+μ_12^(x1)+…+μ_s2^(xs)+γ|＜ηhas infinitely many solutions in primes p_1,p_2,p_3,p_4 and positive integers x_1,…,x_s.

Proportionally Modular Diophantine Inequalities and Their Multiplicity

Let I be an interval of positive rational numbers. Then the set S （I） = T ∩ N, where T is the submonoid of （Q0＋, ＋） generated by T, is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of a Diophantine inequality of the form ax rood b 〈 cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S （I） has a given multiplicity. We also characterize the numerical semigroups associated with these maximal intervals.

Cubic Diophantine Inequalities

Let λ1, λ2,...,λ7 be real numbers satisfying λi≥1. In this paper, we prove there are integers x1,...,x7 such that the inequalities |λ1x13+λ2x23+...+λ7x73|【1 and 0【sum from i=1 to7(λi|xi]3 （λ1λ2…λ7）89814) hold simultaneously.

Results of Diophantine approximation by unlike powers of primes

Let k be an integer with k ≥ 6. Suppose that λ1, λ2, ... , λ5 be nonzero real numbers not all of the same sign, satisfying that λ1/λ2 is irrational, and suppose that η is a real number. In this paper, for any ε 〉 0, we consider the inequality ｜λ1p1 ＋ λ2p2^2 ＋λ3p3^3＋λ4p4^4 ＋ λ5p5^k ＋ η｜ 〈 （maxpj）^-σ（k）＋ε has infinitely many solutions in prime variables P1, P2,...,P5, where σ（k） depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.

The Solubility of Certain Diophantine Inequalities

Let λ1, λ2,…,λs be s non-zero real numbers not all of the same sign and not all in rational ratio, and k be a natural number; let D（k） be the least s for which the inequality |η+sum from j=1 to s（λj njk）|【（maxnj）-δ has infinitely many solutions. In this paper, we give some new estimations of D（k） for k≥6.

Diophantine inequalities over Piatetski-Shapiro primes

Let c>1 and 0<γ<1.We study the solubility of the Diophantine inequality∣p^(c)_(1)+p^(c)_(2)+…+p^(c)_(s)−N∣<(log N)^(−1) in Piatetski-Shapiro primes p_(1),p_(2),…,p_(s) of the form pj=[m^(1/γ)]for some m∈ℕ,and improve the previous results in the cases s=2,3,4.

Values of binary linear forms at prime arguments

We investigate the exceptional set of real numbers not close to some value of a given binary linear form at prime arguments. Let λ1 and λ2 be positive real numbers such that λ1/λ2 is irrational and algebraic. For any （C, c） well-spaced sequence v and δ 〉 0, let E（v, X,δ ） denote the number of ν∈v with v ≤ X for which the inequality ｜λ1P1 ＋ λ2P2 - v｜ 〈 v-δ has no solution in primes P1,P2. It is shown that for any ε 〉 0, we have E（V, X, δ） 〈〈 max（X3/5＋2δ＋ε, X1/3＋4/3δ＋ε）.