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.
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![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.](/uploads/image/z/7115528-56-8.jpg?t=%E6%95%B0%E8%AE%BA%E4%B8%80%E9%A2%98Let+m%2Cn%E2%88%88Z+and+m+odd.Show+that+%282%5Em-1%2C2%5En%2B1%29%3D1+and+%284%5Em-1%2C4%5En%2B1%29%3D1.%E7%BF%BB%E8%AF%91%EF%BC%9Am%2Cn%E6%98%AF%E6%95%B4%E6%95%B0%2C%E4%B8%94m%E6%98%AF%E5%A5%87%E6%95%B0%2C%E6%B1%82%E8%AF%812%5Em-1%E5%92%8C2%5En%2B1%E4%BA%92%E7%B4%A0.%E4%B8%944%5Em-1%E5%92%8C4%5En%2B1%E4%BA%92%E7%B4%A0.)
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