Squared triangular number

Square of a triangular number
triangular numbers that are themselves square

In number theory, the sum of the first n cubes is the square of the nth triangular number. That is,

The same equation may be written more compactly using the mathematical notation for summation:

This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE).

Nicomachus, at the end of Chapter 20 of his Introduction to Arithmetic, pointed out that if one writes a list of the odd numbers, the first is the cube of 1, the sum of the next two is the cube of 2, the sum of the next three is the cube of 3, and so on. He does not go further than this, but from this it follows that the sum of the first n cubes equals the sum of the first odd numbers, that is, the odd numbers from 1 to n(n + 1) − 1. The average of these numbers is obviously , and there are of them, so their sum is
Many early mathematicians have studied and provided proofs of Nicomachus's theorem. claims that "every student of number theory surely must have marveled at this miraculous fact". finds references to the identity not only in the works of Nicomachus in what is now Jordan in the 1st century CE, but also in those of Aryabhata in India in the 5th century, and in those of Al Karaji c. 1000 in Persia. mentions several additional early mathematical works on this formula, by Al Qabisi (10th century Arabia), Gersonides (c. 1300, France), and Nilakantha Somayaji (c. 1500, India); he reproduces Nilakantha's visual proof.

A similar result to Nicomachus's theorem holds for all power sums, namely that odd power sums (sums of odd powers) are a polynomial in triangular numbers. These are called Faulhaber polynomials, of which the sum of cubes is the simplest and most elegant example. However, in no other case is one power sum a square of another. studies more general conditions under which the sum of a consecutive sequence of cubes forms a square. and study polynomial analogues of the square triangular number formula, in which series of polynomials add to the square of another polynomial.