一个素变量丢番图问题

A Diophantine Problem with Prime Variables

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0<σ<2^(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.

Let k and r be positive integers with k≥ 3 and r ≥ψ（k） ＋ 1, where ψ（k） = 2k-1 for 3 ≤ k ≤4, and ψ（k） = ψk（k＋1） for k ≥ 5. Suppose that δand ε are fixed and sufficiently small positive numbers, λ1, λ2,.. , λr＋1 are nonzero real numbers, not all of the same sign 0〈σ〈2 1-2k/r-1 and not all in rational ratios. Then for any real η and 0 〈 σ 〈 τ-1 2 1-2k, it is proved that there exists a positive sequence X →＋∞, such that the inequality ｜λ1p1k＋λ2p2k＋…＋λτpτk＋λτ＋1pτ｜1＋η｜〈）1≤j≤τ＋1maxpj）-σ has 〉〉Xk-τ-τ-1 2 1-2k＋ε prime solutions （p1,p2,… ,Pτ＋1） with （δX）k/1≤pj≤Xk/1（1≤j≤τ） and εX≤pτ＋1≤X. This gives an improvement of an earlier result.