A perfect square is the result of multiplying an integer by itself: \(n = k^2\) for some integer \(k\). The sequence: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
Perfect squares show up everywhere: the Pythagorean theorem equates the squares of triangle side lengths; quadratic equations are solved via the discriminant \(b^2 - 4ac\); the area of a square of side \(k\) is \(k^2\).
In modular arithmetic, quadratic residues — squares mod \(p\) — are central to results like quadratic reciprocity. The differences between consecutive squares form the odd numbers: \((k+1)^2 - k^2 = 2k+1\).