Visual demonstration that the square of a triangular number equals a sum of cubes.

In number theory, the sum of the first n cubes is the square of a triangular number. That is, Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a perfect cube or cube number, is an integer that can be written as the cube (arithmetic) of some other integer. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer. ... A triangular number (so called because it can be arranged into a triangle) is the sum of the n natural numbers from 1 to n. ...

The sequence of numbers formed by this identity is In mathematics, the term identity has several important uses: identity can refer to an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ...

0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, ... (sequence A000537 in OEIS).

These numbers can be viewed as figurate numbers, a four-dimensional hyperpyramidal generalization of the triangular numbers and square pyramidal numbers. As Stein (1971) observes, these numbers also count the number of rectangles with horizontal and vertical sides formed in an n×n grid. For instance, the points of a 4×4 grid can form 36 different rectangles. The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... A figurate number is a number that can be represented as a regular and discrete geometric pattern (e. ... A triangular number (so called because it can be arranged into a triangle) is the sum of the n natural numbers from 1 to n. ... A pyramidal number, or square pyramidal number, is a figurate number that represents a pyramid with a base and four sides. ... Upright square tiling. ...

Stroeker (1995) writes that "every student of number theory surely must have marveled at" this "miraculous fact". While Stroeker's statement may perhaps be a poetic exaggeration, it is true that many mathematicians have studied this equality and have proven it in many different ways. Pengelley (2002) finds references to the identity in several ancient mathematical texts: the works of Nicomachus in what is now Jordan in the first century B.C.E., Aryabhata in India in the fifth century, and Al-Karaji circa 1000 in Persia. Bressoud (2004) mentions several additional early mathematical works on this formula, by Alchabitius (tenth century Arabia), Gersonides (circa 1300 France), and Nilakantha Somayaji (circa 1500 India); he reproduces Nilakantha's visual proof. Nicomachus (Gr. ... Statue of Aryabhata on the grounds of IUCAA, Pune. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ... For other uses of this term see: Persia (disambiguation) The Persian Empire is the name used to refer to a number of historic dynasties that have ruled the country of Persia (Iran). ... Alchabitius was the common European name of a 10th century Arabian astrologer, also known by the transliterated Arabic name Abdelazys. ... Levi ben Gershon (Levi son of Gerson), better known as Gersonides or the Ralbag (1288-1344), was a famous rabbi, philosopher, mathematician and Talmudic commentator. ... Nilakantha Somayaji (नीलकण्ठ सोमयाजि) (1444-1544), from Kerala, was a major mathematician and astronomer. ...

Wheatstone (1854) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers: Charles Wheatstone Sir Charles Wheatstone (February 6, 1802 - October 19, 1875) was the British inventor of many innovations including the English concertina the Stereoscope an early form of microphone the Playfair cipher (named for Lord Playfair, the person who publicized it) He was a major figure in the development of...

1 + 8 + 27 + 64 + 125 + ...

  = (1) + (3 + 5) + (7 + 9 + 11) + (13 + 15 + 17 + 19) + (21 + 23 + 25 + 27 + 29) + ...

  = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 ...

The sum of any set of consecutive odd numbers starting from 1 is a square, and the quantity that is squared is the count of odd numbers in the sum, which is easily seen to be triangular.

In the more recent mathematical literature, Stein (1971) uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also Benjamin et al); he observes that it may also be proved easily (but uninformatively) by induction, and states that Toeplitz (1963) provides "an interesting old Arabic proof". Kanim (2004) provides a purely visual proof, Benjamin and Orrison (2002) provide two additional proofs, and Nelsen (1993) gives seven geometric proofs.

Stroeker (1995) studies more general conditions under which the sum of a consecutive sequence of cubes forms a square. Garrett and Hummel (2004) and Warnaar (2004) study polynomial analogues of the square triangular number formula, in which series of polynomials add to the square of another polynomial.

References