一个二元丢番图不等式(英文)

On a Binary Diophantine Inequality

本文证明了如果1<c<43/36,则二元素变数丢番图不等式|P_1~c+P_2~c-N|<N^(1-43/36c)对几乎所有的正数N是可解的,改进了Laporta的结果1<c<15/14。我们也研究了番图不等式|P^c+P_k^c-N|<ε对于充分大的正数N的可解性问题,这里p是素数,而P_k表示最多有k个素因子的正整数,ε为一充分小的正数。

In this paper we prove that if 1 < c < 43/36,then the Diopantine inequality |P1c + p2c - N| < N1-43/36c is solvable in prime numbers p1 and p2 for 'almost all' N > 0, which improves Laporta's range 1 < c < 15/14. For sufficiently large N, we also investigate the solvable of the Diophantine inequality |pc + Pkc - N|< ε in a prime p and a positive integer Pk with at most k prime factors.