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

$$\sum_{i=1}^{n} i^3 = \Bigl(\sum_{i=1}^{n} i\Bigr)^2.$$

This identity is sometimes called Nicomachus's theorem.

History

Stroeker (1995), writing about Nicomachus's theorem, claims 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 CE, 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.
Numeric values; geometric and probabilistic interpretation

The sequence of squared triangular numbers is

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 number of squares in a square grid is similarly counted by the square pyramidal numbers.

The identity also admits a natural probabilistic interpretation as follows. Let X, Y, Z, W be four integer numbers independently and uniformly chosen at random between 1 and n. Then, the probability that W be not less than any other is equal to the probability that both Y be not less than X and W be not less than Z, that is, $$\scriptstyle \mathbb{P}\left(\{\max(X,Y,Z)\leq W\}\right)=\mathbb{P}\left(\{X\leq Y\} \cap \{Z\leq W\}\right)$$. Indeed, these probabilities are respectively the left and right sides of the Nichomacus identity, normalized over $$n^4$$.
Proofs

Wheatstone (1854) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers:

\begin{align} \sum_{k=1}^n k^3 &= 1 + 8 + 27 + 64 + \cdots + n^3 \\ &= \underbrace{1}_1 + \underbrace{3+5}_2 + \underbrace{7 + 9 + 11}_3 + \underbrace{13 + 15 + 17 + 19}_4 + \cdots + \underbrace{n^2-n+1 + \cdots + n^2+n-1}_n \\ &= \underbrace{1 + 3 + 5 + 7 + 9 + \cdots + n^2 + n - 1}_{1 + 2 + \cdots + n} \\ &= (1 + 2 + \cdots + n)^2 = \left(\sum_{k=1}^n k\right)^2. \end{align}

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. The latter is easily seen to be a count of the form 1+2+3+4+...+n.

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.
Generalizations

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.

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

Benjamin, Arthur T.; Orrison, M. E. (2002). "Two quick combinatorial proofs of $$\scriptstyle \sum k^3 = {n+1\choose 2}^2$$". The College Mathematics Journal 33 (5): 406–408.
Benjamin, Arthur T.; Quinn, Jennifer L.; Wurtz, Calyssa (2006). "Summing cubes by counting rectangles". The College Mathematics Journal 37 (5): 387–389. doi:10.2307/27646391. ISSN 0746-8342. JSTOR 27646391.
Bressoud, David (2004). Calculus before Newton and Leibniz, Part III. AP Central.
Garrett, Kristina C.; Hummel, Kristen (2004). "A combinatorial proof of the sum of q-cubes". Electronic Journal of Combinatorics 11 (1): Research Paper 9. MR2034423.
Kanim, Katherine (2004). "Proofs without Words: The Sum of Cubes—An Extension of Archimedes' Sum of Squares". Mathematics Magazine 77 (4): 298–299. doi:10.2307/3219288. JSTOR 3219288.
Nelsen, Roger B. (1993). Proofs without Words. Cambridge University Press. ISBN 978-0-88385-700-7.
Pengelley, David (2002). "The bridge between continuous and discrete via original sources". Study the Masters: The Abel-Fauvel Conference. National Center for Mathematics Education, Univ. of Gothenburg, Sweden.
Stein, Robert G. (1971). "A combinatorial proof that $$\scriptstyle \sum k^3 = (\sum k)^2$$". Mathematics Magazine (Mathematical Association of America) 44 (3): 161–162. doi:10.2307/2688231. JSTOR 2688231.
Stroeker, R. J. (1995). "On the sum of consecutive cubes being a perfect square". Compositio Mathematica 97 (1–2): 295–307. MR1355130.
Toeplitz, Otto (1963). The Calculus, a Genetic Approach. University of Chicago Press. ISBN 978-0-226-80667-9.
Warnaar, S. Ole (2004). "On the q-analogue of the sum of cubes". Electronic Journal of Combinatorics 11 (1): Note 13. MR2114194.
Wheatstone, C. (1854). "On the formation of powers from arithmetical progressions". Proceedings of the Royal Society of London 7: 145–151. doi:10.1098/rspl.1854.0036.