Definition

A perfect square is a number you get when you multiply a whole number by itself, one that can be arranged as a square of dots. Formally, it is an integer of the form $n^2$ for some non-negative integer $n$; its square root is always a whole number, and perfect squares have an odd number of total factors, since the middle factor $\sqrt{n}$ is shared by its own pair. An integer $n \ge 0$ is a perfect square if and only if every exponent in its prime factorization $n = p_1^{a_1} \cdots p_k^{a_k}$ is even, equivalently if $n$ is a quadratic residue modulo every prime; Fermat's theorem on sums of two squares states that a prime $p$ is a sum of two squares ($p = a^2+b^2$) if and only if $p=2$ or $p \equiv 1 \pmod 4$.