number Theory question i need detailed solution Let p be an

number Theory question! i need detailed solution

Let p be an odd prime and let n reaterthanorequalto 1 be an integer. Show that there is no solution of X^2 = 1 (mod p^ n) other than x = plusminus1 (mod p^ n).

Solution

Suppose x² 1 (mod pⁿ).

Then, x² 1 0 (mod pⁿ).

Hence, (x 1)(x + 1) 0 (mod pⁿ).

Case 1: gcd(p, x 1) = 1

If gcd(p, x 1) = 1, x 1 has an inverse, t modulo p and hence, 0 t(x 1)(x + 1) (mod pⁿ) (x + 1) (mod pⁿ).

Hence, x 1 (mod pⁿ).

Case 2: gcd(p, x 1) > 1 p is prime, therefore, if gcd(p, x 1) > 1, gcd(p, x 1) = p and thus x 1 = kp. Hence, x = kp + 1. i.e. x 1 (mod pⁿ).

Therefore x²1 is solvable only at x= -1, 1