解不定方程3^x=2^y+1

Question

解不定方程3^x=2^y+1

Answer 1

显然 x > 0,
当y = 1时,y = 1,x = 1.
以下 y >1,
3^x = 2^y + 1 (mod4) => （-1)^x = 1(mod4) => x = 0 mode(2)
x = 2m,m > 0
(3^m+1)(3^m-1) = 2^y
所以 3^m+1,3^m-1 都是 2的幂次,并且：(3^m+1)-(3^m-1)=2.
所以 必须有：3^m+1=4,3^m-1 = 2 => m = 1.=> x = 2,y = 3.
所以
1.y = 1,x = 1；
2.x = 2,y = 3.