With this knowledge if we lift the result from primes to composites using CRT (Chinese Remainder Theorem) we can observe that the equation x² ≡ a mod n has 2^k roots when n is odd and when n is even it has 2^k-1, 2^k and 2^k+1 roots respectively for e₁=1, e₁=2 and e₁≥ 2.
Now we introduce to the notion of Legendre Symbol of an element . It is denoted by and is defined as follows:
= =
depending on whether a is a quadratic residue or non-quadratic residue from Euler’s Criterion. 

Theorem 3: For every odd prime p we have   = (-1) =  depending on p ≡1 mod 4 and p ≡ 3 mod 4 respectively.

Theorem 4:  For every odd prime p we have   = = +1 (if p ≡ ±1 mod 8) and = -1 (if p ≡ ±3 mod 8).
Proof:  Consider the following congruences :
p-1 ≡ 1(-1)¹  mod p, 2 ≡ 2(-1)² mod p, p -3 ≡ 3(-1)³ mod p, 4 ≡ (-1)⁴ mod p, …, r ≡  mod p. Here r is either p – (p-1)/2 or (p-1)/2. If we multiply these congruences and observing the fact that the number on the left of each congruence is even, we obtain:

2.4.6 … (p-1) ≡ ! (-1) mod p.
Thus we have ! ≡ !(-1) mod p. Since !≡ 0 mod p we have established the first equality since   = .