一个素数形式为p=x^(2)+y^(2)+1的的丢番图不等式

A Diophantine inequality with one prime of the form p=x^(2)+y^(2)+1

本文的目的是解决一个素数为特殊形式的六元丢番图不等式.更准确地说,令1<c<1831/1264是一个固定的实数,N是一个充分大的正数并且ε表示一个较小的正常数.我们证明了丢番图不等式|p_(1)^(c)+p_(2)^(c)+…+p_(6)^(c)-N|<ε在素变量p_(1),p_(2),…,p_(6)上有解,并且使得p_(1)=x^(2)+y^(2)+1,这时x和y为整数.

Our aim of this paper is to solve the Diophantine inequality with six prime numbers of a special form. More precisely, let 1<c<1831/1264 be a fixed real number, N be a sufficiently large positive number and ε denote a small positive constant. We prove that the Diophantine inequality |p_(1)^(c)+p_(2)^(c)+…+p_(6)^(c)-N|<ε is solvable in prime variables p_(1), p_(2),…, p_(6) such that p_(1) = x^(2)+ y^(2)+1 with integers x and y.