# Difference between revisions of "Perfect square"

An integer $n$ is said to be a perfect square if there is an integer $m$ so that $m^2=n$. The first few perfect squares are 0, 1, 4, 9, 16, 25, 36.

The sum of the first $n$ square numbers (not including 0) is $\frac{n(n+1)(2n+1)}{6}$

An integer $n$ is a perfect square iff it is a quadratic residue modulo all but finitely primes.

## Perfect Square Trinomials

Another type of perfect square is an equation that is a perfect square trinomial. Take for example

$(x+a)^2=x^2+2xa+a^2$.

Perfect square trinomials are a type of quadratic equation that have 3 terms and contain 1 unique root.

For any quadratic equation in the form $ax^2+bx+c$, it is a perfect square trinomial iff $b=a\sqrt{c}$.