Cube numbers

Cubes had a geometric meaning before they had an algebraic one. For Greek mathematicians n³ was a solid, and the famous problem of doubling the cube — constructing a cube of twice the volume of a given one, using only compass and straightedge — stood unresolved for two thousand years. Legend places its origin at Delos, where an oracle supposedly instructed the islanders to double the size of an altar to end a plague. The problem is impossible, which Pierre Wantzel proved in 1837 by showing that the construction would require the cube root of two, a number not obtainable by the permitted operations.

The most striking fact about cubes is a statement about their sums. Add the first n cubes and you always get a perfect square — specifically the square of the nth triangular number. So 1 + 8 + 27 + 64 = 100 = 10², and 10 is the fourth triangular number. The result is called Nicomachus's theorem after Nicomachus of Gerasa, whose Introduction to Arithmetic of around 100 CE records it, and it has one of the richest collections of visual proofs of any identity in elementary mathematics.

Cubes also gave number theory one of its longest-running stories. Fermat's Last Theorem for the exponent 3 — that no positive cubes sum to another cube — was the first case proved beyond the Pythagorean one, by Leonhard Euler in the 1770s, though his argument contained a gap later repaired. The general theorem waited until Andrew Wiles in 1994.

Then there is 1729, which Srinivasa Ramanujan identified from a hospital bed as the smallest number expressible as a sum of two cubes in two different ways: 1³ + 12³ and 9³ + 10³. G. H. Hardy had remarked that the taxi number seemed dull.

Key properties

• C(n) = n³, and the difference between consecutive cubes is 3n² + 3n + 1.
• The sum of the first n cubes is (n(n+1)/2)² — the square of the nth triangular number.
• Every cube is congruent to 0, 1 or −1 modulo 9, which rules out many candidate equations immediately.
• Unlike squares, cubes preserve sign: the cube of a negative number is negative, so every real number has exactly one real cube root.
• Every integer is the sum of at most nine positive cubes, and all but finitely many need at most seven.
• 1729 is the smallest number expressible as a sum of two positive cubes in two distinct ways: 1³ + 12³ = 9³ + 10³.
• No three positive cubes sum to a cube — Fermat’s Last Theorem for exponent 3, first proved by Euler.

Where they turn up

• Volume scaling: doubling a linear dimension multiplies volume by eight, which is why large animals cannot simply be scaled-up small ones.
• Cubic equations, whose general solution by Cardano and Tartaglia in the sixteenth century forced the first serious use of complex numbers.
• Cubic splines and Bézier curves, the cubic polynomials behind most vector graphics and font outlines.
• The Rubik’s cube and related puzzles, where the 3×3×3 structure makes 27 the governing count.
• The Delian problem of doubling the cube, one of the three classical construction problems proved impossible in the nineteenth century.