一个素变量丢番图问题

设k和r是满足k≥3及r≥Ψ(k)+1的正整数,这里当3≤k≤4时,Ψ(k)=2~(k-1);而当k≥5时,Ψ(k)=1/2k(k+1).假定δ和ε是给定的足够小的正数,λ_1,λ_2,…,λ_(r+1)是不全同号且两两之比不全为有理数的非零实数.对于任意实数η与0σ2~(1-2k)/r-1,证明了:存在一个正数序列X→+∞,使得不等式|λ_1p_1~k+λ_2p_2~k+···+λ_rp_r~k+λ_(r+1)p_(r+1)+η|(max(1≤j≤r+1)p_j)~(-σ)有》■X~(■-(2~(1-2k))/(r-1)+ε组素数解(p_1,p_2,…,p_(r+1)),这里(δX)~(1/k)≤p_j≤X~(1/k)(1≤j≤r)及δX≤p_(r+1)≤X.这改进了之前的结果.

A Diophantine Problem with Prime Variables

Let $k$ and $r$ be positive integers with $k\geq 3$ and $r\geq \psi(k)+1$, where $\psi(k)=2^{k-1}$ for $3\leq k\leq 4$, and $\psi(k)=\frac{1}{2}k(k+1)$ for $k\geq 5$. Suppose that $\delta$ and $\varepsilon$ are fixed and sufficiently small positive numbers, $\lambda_1, \lambda_2, \cdots, \lambda_{r+1}$ are nonzero real numbers, not all of the same sign and not all in rational ratios. Then for any real $\eta$ and $0<\sigma<\frac{2^{1-2k}}{r-1}$, it is proved that there exists a positive sequence $X\rightarrow +\infty$, such that the inequality $$|\lambda_1p_1^k+\lambda_2p_2^k+\cdots+\lambda_rp_r^k+\lambda_{r+1}p_{r+1}+\eta|<\big(\max_{1\leq j\leq r+1}p_j\big)^{-\sigma}$$ has $\gg X^{\frac{r}{k}-\frac{2^{1-2k}}{r-1}+\varepsilon}$ prime solutions $(p_1, p_2, \cdots, p_{r+1})$ with $(\delta X)^\frac{1}{k}\leq p_j\leq X^\frac{1}{k}\,(1\leq j\leq r)$ and $\delta X\leq p_{r+1}\leq X$. This gives an improvement of an earlier result.