【学习笔记】数论、数学—补充证明 (3)

【学习笔记】数论、数学—补充证明 (3)

\(\text{结论:}\)

\[\exists x\in N^{*},a^x=1(\bmod m) \iff \gcd(a,m)=1 \]


\(\text{证明(右推左):}\)

\(\text{由欧拉定理可知当}\ x=\varphi(m)\ \text{时该同余方程成立。}\)

\(\text{证毕。}\)


\(\text{证明(左推右):}\)

\(\text{易知}\) \(\exists x\in N^{*},\gcd(a^x,m)=\gcd(a^x,a^x\mod m)=\gcd(a^x,1)=1\)

\(\text{假设}\) \(\gcd(a,m)\neq1\)\(\text{则}\) \(\forall k\in N^{*}, \gcd(a^k,m)\neq1\)

\(\text{所以}\) \(\forall k\in N^{*}, \gcd(a^k,a^k\mod m)\neq 1,\) \(\text{与前者}\) \(\exists x\in N^{*},\gcd(a^x,m)=1\) \(\text{矛盾}\)

\(\text{所以必有}\) \(\gcd(a,m)=1\)

\(\text{证毕。}\)

posted @ 2020-07-01 16:24  辰星凌  阅读(40)  评论(0编辑  收藏