Perfect square

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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 (starting with $1$ ) 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

A type of perfect square is an equation that is a perfect square trinomial. 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 $b=a\sqrt{c}$ .