关于Diophantine方程x+y+xy=2^(p-1)

On the Diophantine Equatin x+y+xy=2^(p-1)

设p是素数,对于非负整数k,设F(k)=22k+1是第k个Fermat数,本文证明了:方程x+y+xy=2p-1没有正整数解(x,y)的充要条件是P=2或者P=F(k)且F(2k)也是素数.

Letp be aprime. For any nonnegative integer k, let F（k） = 2^2k ＋ 1 be the k -th Fermat number. In this paper we prove that equation x ＋ y ＋ xy = 29-1 has no positive integer solution （x ,y） if and only if either p = 2 orp = F（k） and F（2k） is also aprime.