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.
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
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.
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
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 λ1,λ2,λ3,λ4 be non-zero real numbers, not all of the same sign, w real. Suppose that the ratios λ1/λ2,λ1/λ3 are irrational and algebraic. Then there are infinitely many solutions in primes pj, j = 1, 2,3,4...Let λ1,λ2,λ3,λ4 be non-zero real numbers, not all of the same sign, w real. Suppose that the ratios λ1/λ2,λ1/λ3 are irrational and algebraic. Then there are infinitely many solutions in primes pj, j = 1, 2,3,4, to the inequality |λ1p1 +λ2p^2/2 +λ3p^2/3+|λ4p^2/4+■(max{p1,p^2/2,p^2/3,p^2/4})-^5/64. This improves the earlier result.展开更多
基金Acknowledgements The author was grateful to his supervisor, Professor Hongze Li, for his guidance and support. The author would like to thank Quanwu Mu for his warm heart.He gave talks on diophantine inequalities to the author individually and provided helpful discussion. This work was supported by the National Natural Science Foundation of China （Grant No. 11271249） and Specialized Research Fund for the Doctoral Program of Higher Education （No. 20120073110059）.
文摘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.
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.
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.
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.
基金The authors thank the referees for their time and comments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11871193, 11471112)the Key Research Project of Henan Province Higher Education (No. 17A110009).
文摘Let λ1,λ2,λ3,λ4 be non-zero real numbers, not all of the same sign, w real. Suppose that the ratios λ1/λ2,λ1/λ3 are irrational and algebraic. Then there are infinitely many solutions in primes pj, j = 1, 2,3,4, to the inequality |λ1p1 +λ2p^2/2 +λ3p^2/3+|λ4p^2/4+■(max{p1,p^2/2,p^2/3,p^2/4})-^5/64. This improves the earlier result.
