Let p be a prime number and gcdp n 1 Define an equivalence

Let p be a prime number and gcd(p, n) = 1. Define an equivalence relation on Z_p as follows: x ~ y if and only if n^r x = n^t y for some r, t greaterthanorequalto 0. Let m be the number of equivalence classes of this equivalence relation. Prove that m - 1 is a divisor of p - 1.

Solution

solution -:

According to the Wilson\'s theorem, the natural number p is prime if and only if (p1)!1(modp), hence p|(p1)!+1(p1)!+1=pkpk+(1)(p1)!=1(p,(p1)!)=1