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, λ1, λ2, λ3是非零实数, 不全同号, 并且λ1/λ2 是无理数, 则不等式 |λ1p1+λ2p2+λ3p3k+η|<(max pj)-σ 有无穷多组素数解 p1, p2, p3, 这里 σ 满足: 当 2≦k≦ 3 时, 0<σ<1/2(2k+1+1);当 4≦k≦ 5 时, 0<σ<5/(6k2k); 当 k≧ 6 时, 0<σ<20/(21k2k).
Abstract
Let k be an integer with k ≥ 2 and η be any real number. Suppose that λ1, λ2, λ3 are nonzero real numbers, not all of the same sign and λ1/λ2 is irrational. It is proved that the inequality |λ1p1+λ2p2+λ3p3k +η| < (max pj)-σ has infinitely many solutions in prime variables p1, p2, p3, where 0 < σ < 1/2(2k+1+1) for 2 ≤ k ≤ 3, 0 < σ < 5/(6k2k) for 4 ≤ k ≤ 5, and 0 < σ < 20/(21kk) for k ≥ 6.
关键词
丢番图不等式 /
混合幂 /
Davenport&mdash /
Heilbronn 方法
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Key words
Diophantine inequality /
mixed power /
Davenport-Heilbronn method
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参考文献
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脚注
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