Question

在Z7中,证明:[5]^7=[2]^7+[3]^7,[5]^7=[5]

Answer 1

Z7就是模7的余数的域,在此域中[7]=[0],[8]=[1],[9]=[2],.

[5]^7=([2]+[3])^7=[2]^7+∑C(7,k) [2]^k [3]^(7-k) +[3]^7

其中求和k从1到6 ,注意C(7,k)均是7的倍数,所以在Z7中C(7,k)=0

[5]^7=[2]^7+[3]^7

[5]^7=[5][5]^6=[5] [25]^3=[5][4]^3=[5][16]^3=[5][2]^3=[5][8]=[40]=[5]