View source for Squared triangular number

{{short description|Square of a triangular number}}
{{For|triangular numbers that are themselves square|square triangular number}}

[[File:Nicomachus_theorem_3D.svg|thumb|upright=1.2|right|A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. From {{harvtxt|Gulley|2010}}.

The {{mvar|n}}th coloured region shows {{mvar|n}} squares of dimension {{mvar|n}} by {{mvar|n}} (the rectangle is 1 evenly divided square), hence the area of the {{mvar|n}}th region is {{mvar|n}} times {{math|''n'' &times; ''n''}}.]]

In [[number theory]], the sum of the first {{mvar|n}} [[Cube (algebra)|cube]]s is the [[Square number|square]] of the {{mvar|n}}th [[triangular number]]. That is,
:<math>1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2.</math>
The same equation may be written more compactly using the mathematical notation for [[summation]]:
:<math>\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2.</math>

This [[Identity (mathematics)|identity]] is sometimes called '''Nicomachus's theorem''', after [[Nicomachus|Nicomachus of Gerasa]] ({{circa|60}} – {{circa|120 CE}}).

==History==
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 <math>n</math> cubes equals the sum of the first <math>\tfrac{n(n+1)}{2}</math> odd numbers, that is, the odd numbers from 1 to <math>n(n+1)-1</math>. The average of these numbers is obviously <math>\tfrac{n(n+1)}{2}</math>, and there are <math>\tfrac{n(n+1)}{2}</math> of them, so their sum is <math>\left(\tfrac{n(n+1)}{2}\right)^2</math>.

Many early mathematicians have studied and provided proofs of Nicomachus's theorem. {{harvtxt|Stroeker|1995}} claims that "every student of number theory surely must have marveled at this miraculous fact".{{sfnp|Stroeker|1995}} {{harvtxt|Pengelley|2002}} 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]] {{circa|1000}} in [[Persia]].{{sfnp|Pengelley|2002}} {{harvtxt|Bressoud|2004}} mentions several additional early mathematical works on this formula, by [[Al-Qabisi]] (10th century Arabia), [[Gersonides]] ({{circa|1300}}, France), and [[Nilakantha Somayaji]] ({{circa|1500}}, India); he reproduces Nilakantha's visual proof.{{sfnp|Bressoud|2004}}

==Numeric values;  geometric and probabilistic interpretation==
[[File:grid_rectangle_count_puzzle.svg|thumb|upright=1.2|All 36 ({{nowrap|1== (1 + 2 + 3)<sup>2</sup>}} = {{nowrap|1<sup>3</sup> + 2<sup>3</sup> + 3<sup>3</sup>}}) rectangles, including [[square_pyramidal_number#Geometric_enumeration|14 ({{nowrap|1== 1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup>}}) squares]] (red), in a 3&nbsp;×&nbsp;3 square {{nowrap|(4&nbsp;×&nbsp;4 vertex) grid}}]]
The sequence of squared triangular numbers is
{{Block indent|left=1.6|{{math|[[0 (number)|0]], [[1 (number)|1]], [[9 (number)|9]], [[36 (number)|36]], [[100 (number)|100]], 225,}} {{math|441, 784, 1296, 2025, 3025, 4356, 6084, 8281,}} ... {{OEIS|id=A000537}}.}}
These numbers can be viewed as [[figurate number]]s, a four-dimensional hyperpyramidal generalization of the [[triangular number]]s and [[square pyramidal number]]s.

As {{harvtxt|Stein|1971}} observes, these numbers also count the number of rectangles with horizontal and vertical sides formed in an <math>n\times n</math> [[Square lattice|grid]]. For instance, the points of a <math>4\times 4</math> grid (or a square made up of three smaller squares on a side) can form 36 different rectangles. The number of squares in a square grid is similarly counted by the square pyramidal numbers.{{sfnp|Stein|1971}}

The identity also admits a natural probabilistic interpretation as follows. Let <math>X,Y,Z,W</math> be four integer numbers independently and uniformly chosen at random between 1 and <math>n</math>. Then, the probability that <math>W</math> is the largest of the four numbers equals the probability that <math>Y</math> is at least as large as <math>X</math> and that <math>W</math> is at least as large as <math>Z</math>. That is, <math display=block>\Pr[\max(X,Y,Z)\le W]=\Pr[X\le Y\wedge Z\le W].</math> For any particular value of <math>W</math>, the combinations of <math>X</math>, <math>Y</math>, and <math>Z</math> that make <math>W</math> largest form a cube <math>1\le X,Y,Z\le n</math> so (adding the size of this cube over all choices of <math>W</math>}) the number of combinations of <math>X,Y,Z,W</math> for which <math>W</math> is largest is a sum of cubes, the left hand side of the Nichomachus identity. The sets of pairs <math>(X,Y)</math> with <math>X\le Y</math> and of pairs <math>(Z,W)</math> with <math>Z\le W</math> form isosceles right triangles, and the set counted by the right hand side of the equation of probabilities is the [[Cartesian product]] of these two triangles, so its size is the square of a triangular number on the right hand side of the Nichomachus identity. The probabilities themselves are respectively the left and right sides of the Nichomachus identity, normalized to make probabilities by dividing both sides {{nowrap|by <math>n^4</math>.}}{{citation needed|date=July 2021}}

==Proofs==
{{harvs|txt|first=Charles|last=Wheatstone|authorlink=Charles Wheatstone|year=1854}} gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. He begins by giving the identity 
<math display=block>n^3 = \underbrace{\left(n^2-n+1\right) + \left(n^2-n+1+2\right) + \left(n^2-n+1+4\right)+ \cdots + \left(n^2+n-1\right)}_{n \text{ consecutive odd numbers}}.</math>
That identity is related to [[triangular numbers]] <math>T_n</math> in the following way:
<math display=block>n^3 =\sum _{k=T_{n-1}+1}^{T_{n}} (2 k-1),</math>
and thus the summands forming <math>n^3</math> start off just after those forming all previous values <math>1^3</math> up to <math>(n-1)^3</math>. Applying this property, along with another well-known identity:
<math display=block>n^2 = \sum_{k=1}^n (2k-1),</math>
produces the following derivation:{{sfnp|Wheatstone|1854}}
<math display=block>\begin{align}
\sum_{k=1}^n k^3 &= 1 + 8 + 27 + 64 + \cdots + n^3 \\
&= \underbrace{1}_{1^3} + \underbrace{3+5}_{2^3} + \underbrace{7 + 9 + 11}_{3^3} + \underbrace{13 + 15 + 17 + 19}_{4^3} + \cdots + \underbrace{\left(n^2-n+1\right) + \cdots + \left(n^2+n-1\right)}_{n^3} \\
&= \underbrace{\underbrace{\underbrace{\underbrace{1}_{1^2} + 3}_{2^2} + 5}_{3^2} + \cdots + \left(n^2 + n - 1\right)}_{\left( \frac{n^{2}+n}{2} \right)^{2}} \\
&= (1 + 2 + \cdots + n)^2 \\
&= \left(\sum_{k=1}^n k\right)^2.
\end{align}</math>

{{harvtxt|Row|1893}} obtains another proof by summing the numbers in a square [[multiplication table]] in two different ways. The sum of the {{mvar|i}}th row is {{mvar|i}} times a triangular number, from which it follows that the sum of all the rows is the square of a triangular number. Alternatively, one can decompose the table into a sequence of nested [[Gnomon (figure)|gnomon]]s, each consisting of the products in which the larger of the two terms is some fixed value. The sum within each gnomon is a cube, so the sum of the whole table is a sum of cubes.{{sfnp|Row|1893}}

[[File:Sum of cubes2.png|thumb|right|upright=1.6|Visual demonstration that the square of a triangular number equals a sum of cubes.]]
In the more recent mathematical literature, {{harvtxt|Edmonds|1957}} provides a proof using [[summation by parts]].{{sfnp|Edmonds|1957}} {{harvtxt|Stein|1971}} uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity.<ref>{{harvtxt|Stein|1971}}; see also {{harvnb|Benjamin|Quinn|Wurtz|2006}}</ref> Stein observes that it may also be proved easily (but uninformatively) by induction, and states that {{harvtxt|Toeplitz|1963}} provides "an interesting old Arabic proof".{{sfnp|Stein|1971}} {{harvtxt|Kanim|2004}} provides a purely visual proof,{{sfnp|Kanim|2004}} {{harvtxt|Benjamin|Orrison|2002}} provide two additional proofs,{{sfnp|Benjamin|Orrison|2002}} and {{harvtxt|Nelsen|1993}} gives seven geometric proofs.{{sfnp|Nelsen|1993}}

==Generalizations==
A similar result to Nicomachus's theorem holds for all [[Faulhaber's formula|power sums]], namely that odd power sums (sums of odd powers) are a polynomial in triangular numbers.
These are called [[Faulhaber's formula#Faulhaber polynomials|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.{{sfnp|Edmonds|1957}}

{{harvtxt|Stroeker|1995}} studies more general conditions under which the sum of a consecutive sequence of cubes forms a square.{{sfnp|Stroeker|1995}} {{harvtxt|Garrett|Hummel|2004}} and {{harvtxt|Warnaar|2004}} study polynomial analogues of the square triangular number formula, in which series of polynomials add to the square of another polynomial.<ref>{{harvtxt|Garrett|Hummel|2004}}; {{harvtxt|Warnaar|2004}}</ref>

==Notes==
{{Reflist|30em}}

==References==
{{refbegin|30em}}
*{{citation
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 | journal = [[College Mathematics Journal]]
 | year = 2002
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*{{citation
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*{{citation
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*{{citation
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 | jstor = 3609189
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*{{citation
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*{{citation
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 | date = March 4, 2010
 | publisher = Matlab Central}}.
*{{citation
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 | jstor = 3219288
 | journal = [[Mathematics Magazine]]
 | volume = 77
 | issue = 4
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 | pages = 298–299}}.
*{{citation
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 | publisher = Cambridge University Press
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*{{citation
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*{{citation|first=T. Sundara|last=Row|title= Geometric Exercises in Paper Folding|title-link= Geometric Exercises in Paper Folding|year=1893|location=Madras|publisher=Addison|at=[https://archive.org/details/geometricexerci00raogoog/page/n61/mode/2up pp. 47–48]}}.
*{{citation
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 | journal = [[Mathematics Magazine]]
 | volume = 44
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 | year = 1971
 | jstor = 2688231}}.
*{{citation
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*{{citation
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*{{citation
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*{{citation
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{{refend}}

==External links==
*{{mathworld|urlname=NicomachussTheorem|title=Nicomachus's theorem|mode=cs2}}
*[http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32 A visual proof of Nicomachus's theorem] {{Webarchive|url=https://web.archive.org/web/20191019215037/http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32 |date=2019-10-19 }}

{{Figurate numbers}}
{{Classes of natural numbers}}

[[Category:Number theory]]
[[Category:Integer sequences]]
[[Category:Algebraic identities]]
[[Category:Articles containing proofs]]
[[Category:Proof without words]]