Square numbers

Square numbers are the oldest idea in this corner of mathematics, and the name is literal rather than metaphorical. The Pythagoreans of the sixth and fifth centuries BCE arranged pebbles into shapes and classified numbers by the shapes they made; a number was square if its pebbles filled a square. The practice gave figurate numbers their name and gave Greek arithmetic its characteristic geometric flavour.

The arrangement makes one result immediately visible. To grow a square from side n to side n+1, you add an L-shaped border along two edges — the Greeks called this a gnomon, after the upright rod of a sundial. The gnomon added at each step contains 1, then 3, then 5, then 7 dots, which is to say that the sum of the first n odd numbers is exactly n². That is a proof you can see rather than calculate, and it is still the standard way the identity is introduced.

Squares also carry the discovery that broke Pythagorean metaphysics. The school held that all magnitudes were ratios of whole numbers, and the diagonal of a unit square refuted it: no fraction squares to 2. The proof is a parity argument on squares, and the tradition — probably legendary — attributes the discovery to Hippasus of Metapontum and his drowning to the consequences.

Squares of integers have a further property that shaped number theory. Fermat's theorem on sums of two squares states that an odd prime is the sum of two squares exactly when it leaves remainder 1 on division by 4; Lagrange's four-square theorem, proved in 1770, shows that four squares always suffice for any positive integer whatsoever.

Key properties

• S(n) = n², and S(n) − S(n−1) = 2n − 1, so consecutive differences are the odd numbers.
• The sum of the first n odd numbers equals n² — the gnomon identity, visible directly in the dot arrangement.
• A square number ends in 0, 1, 4, 5, 6 or 9 in base 10; it can never end in 2, 3, 7 or 8.
• Every square is congruent to 0 or 1 modulo 4, which is the basis of many impossibility proofs.
• A positive integer has an odd number of divisors precisely when it is a perfect square.
• Lagrange’s four-square theorem: every positive integer is the sum of at most four perfect squares.
• Squares and triangular numbers overlap in the square triangular numbers — 1, 36, 1225, 41616 — which are infinitely many but sparse.

Where they turn up

• The inverse-square law governs gravity, electrostatic force and the intensity of light and sound with distance.
• Algorithmic complexity: an O(n²) nested loop is the classic contrast case against O(n log n) sorting.
• Chessboards, pixel grids and tiling problems, where the count of cells is a square by construction.
• Standard deviation and least-squares regression, which square deviations so that positive and negative errors cannot cancel.
• The Pythagorean theorem, which is a statement about the areas of three squares.