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
本文证明了如果λ_1,λ_2,…,λ_8为不全同号的非零实数,其中λ_1/λ_2为无理数,则对任意实数κ及0<σ<1/8,不等式|λ_1x_1~2+λ_2x_2~2+sum ( λ_ix_i^4+κ) from i=3 to 8|<(max 1≤i≤8 xi)^(-σ)有无穷多组整数解(x_1,x_2,...本文证明了如果λ_1,λ_2,…,λ_8为不全同号的非零实数,其中λ_1/λ_2为无理数,则对任意实数κ及0<σ<1/8,不等式|λ_1x_1~2+λ_2x_2~2+sum ( λ_ix_i^4+κ) from i=3 to 8|<(max 1≤i≤8 xi)^(-σ)有无穷多组整数解(x_1,x_2,…,x_8).展开更多
Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9,...Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9, or b=c=7, then R b,c (n)》n 5/6+1/b+1/c for all sufficiently large n.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.
基金Project supported in part by the Fundamental Research Funds for the Central Universities(No. 2010HGBZ0603)by NSFC(No.11071186 and No.11201107)
文摘本文证明了如果λ_1,λ_2,…,λ_8为不全同号的非零实数,其中λ_1/λ_2为无理数,则对任意实数κ及0<σ<1/8,不等式|λ_1x_1~2+λ_2x_2~2+sum ( λ_ix_i^4+κ) from i=3 to 8|<(max 1≤i≤8 xi)^(-σ)有无穷多组整数解(x_1,x_2,…,x_8).
文摘Let R b,c (n) denote the number of representations of n as the sum of one square, four cubes, one b-th power and one c-th power of natural numbers. It is shown that if b=4, 4 c 35, or b=5, 5 c 13, or b=6, 6 c 9, or b=c=7, then R b,c (n)》n 5/6+1/b+1/c for all sufficiently large n.
