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^...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.展开更多
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.
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.
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...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.展开更多
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 ≤...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.展开更多
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_...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.展开更多
基金the National Natural Science Foundation of China(10671056)
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China(10171076)Supported by the Scientific and Technical Committee Foundation of Shanghai(03JC14027)
文摘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.
基金Supported by the National Natural Science Foundation of China(11201107,11271283,11501435)Supported by the Natural Science Foundation of Anhui Province(1208085QA01)
文摘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.
文摘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.
文摘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.
