数论一题Let m,n∈Z and m odd.Show that (2^m-1,2^n+1)=1 and (4^m-1,4^n+1)=1.翻译：m,n是整数,且m是奇数,求证2^m-1和2^n+1互素.且4^m-1和4^n+1互素.

数论一题Let m,n∈Z and m odd.Show that (2^m-1,2^n+1)=1 and (4^m-1,4^n+1)=1.翻译：m,n是整数,且m是奇数,求证2^m-1和2^n+1互素.且4^m-1和4^n+1互素.
数论一题
Let m,n∈Z and m odd.Show that (2^m-1,2^n+1)=1 and (4^m-1,4^n+1)=1.
翻译：m,n是整数,且m是奇数,求证2^m-1和2^n+1互素.且4^m-1和4^n+1互素.

数论一题Let m,n∈Z and m odd.Show that (2^m-1,2^n+1)=1 and (4^m-1,4^n+1)=1.翻译：m,n是整数,且m是奇数,求证2^m-1和2^n+1互素.且4^m-1和4^n+1互素.
设(2^m-1,2^n+1)=d
所以1=(2^m)^n=(2^n)^m=(-1)^m=-1 (mod d)(此式中等号全为同余符号)
所以d=1或2
又显然d为奇数,所以d=1
得证
后半部分同理可得