Nicomachus's Theorem

Nicomachus's theorem states that the sum of the first positive cubic numbers equals the square of the sum of the first positive integers.

Nicomachus (1926, Ch. 20) noted that when the odd numbers are grouped in blocks of lengths 1, 2, 3, ..., i.e., 1; 3+5; 7+9+11; 13+15+17+19; ..., the th cubic number is a sum of consecutive odd numbers, for example

			(1)
			(2)
			(3)
			(4)

and so on (Boyer and Merzbach 1991, p. 160).

For positive integer , this set of identities follows from

(5)

When coupled with the early Pythagorean recognition that the sum of the first odd numbers is (Boyer and Merzbach 1991, p. 160), this leads to the case of Faulhaber's formula

			(6)
			(7)
			(8)

that is sometimes known as Nicomachus's theorem. For , 2, ..., the first few values are 1, 9, 36, 100, 225, 441, 784, 1296, 2025, ... (OEIS A000537), which are squared triangular numbers and need not themselves be triangular numbers. The values 1 and 36 are also square triangular numbers, whereas 9 is not.

Plugging in gives 3025.