An integer is a quadratic residue with respect to prime if for some integer .

Given a prime and an integer , how fast can we decide whether is a quadratic residue modulo ? As it turns out (courtesy to Euler), pretty fast.

Statement

Given a prime and an integer ,

(First Part) If , it means is a quadratic residue module .

(Second Part) If , it means is a not a quadratic residue module .

One might wonder: what about other possible values of modulo ? As it turns out, modulo can have only two possible values , as shown in the proof of the second part below.

Proof of the First Part

Recall that according to Fermat’s Little Theorem, if integers and are coprime (that is, and do not have any common divisor except 1), the following is true:

, or equivalently

Now, observe that just like , we can write Equation (2) as

Now, since is prime, if the product of two numbers module is zero modulo , at least one of them must be zero modulo . (If were not prime, the product of the factors of would be , and hence the above argument would not hold. Moreover, if is prime, product of two non-zero integers cannot be a multiple of since is coprime to .) Hence, according to this argument, at least one of the factors on the left hand side of Equation (3) (that is, either or ) must be congruent to .

Now suppose for some integer . That is, we are assuming that is a quadratic residue modulo . Then, it follows that

Hence, if is a quadratic residue modulo , the first factor of Equation (3) becomes zero, and thus Equation (3) is satisfied. Equation (4) is the proof of the first part of the statement.

Proof of the Second Part

Now, when is not a quadratic residue modulo , we can observe that

where is an integer such that . The two candidate values for are therefore $\latex \pm 1$. However, cannot be , since it would imply which is identical to Equation (4) which , in turn, would imply is a quadratic residue modulo , contradicting our assumption that is not a quadratic residue modulo . Therefore, , which completes the proof of the second part of the statement.