Diophantine approximation with one prime and three squares of primes
Diophantine approximation with one prime and three squares of primes
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DOI：
10.1016/j.jnt.2017.04.013
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发表时间：
2017-11
影响因子：
0.7
通讯作者：
Weiping Li;Tianze Wang
中科院分区：
文献类型：
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作者：
Weiping Li;Tianze Wang
设λ 1，.，λ 4是非零的真实的数，不全是同号的，λ 1/λ 2是无理数，λ 2是真实的数.我们证明，对于任何ε> 0，不等式|λ 1 p 1+ λ 2 p 2 2+ λ 3 p 3 2+ λ 4 p 4 2+ λ 4 p| ≤（max <${p 1，p 2 2，p 3 2，p 4 2}）− 1 14+ ε在素变量p 1，...，p 4中有无穷多个解。如果我们进一步假设λ 1/λ 3也是无理数，而λ 1/λ 2，λ 1/λ 3都是代数数，那么我们可以用− 3 40+ ε来代替指数。
Abstract Let λ 1,…, λ 4 be non-zero real numbers, not all of the same sign, with λ 1/λ 2 irrational, and ϖ be a real number. We prove that for any ε> 0, the inequality| λ 1 p 1+ λ 2 p 2 2+ λ 3 p 3 2+ λ 4 p 4 2+ ϖ|≤(max{p 1, p 2 2, p 3 2, p 4 2})− 1 14+ ε has infinitely many solutions in prime variables p 1,…, p 4. If we further assume that λ 1/λ 3 is also irrational and λ 1/λ 2, λ 1/λ 3 are both algebraic, then we may replace the exponent by− 3 40+ ε.