|
|
| Line 1: |
Line 1: |
| {{For|triangular numbers that are themselves square|square triangular number}}
| | Hi there, I am Alyson Boon even though it is not the name on my beginning certificate. To perform domino is some thing I truly appreciate performing. I've usually loved living in Kentucky but now I'm considering other choices. Distributing manufacturing is exactly where her main earnings arrives from.<br><br>Here is my web blog ... clairvoyant psychic ([http://appin.co.kr/board_Zqtv22/688025 http://appin.co.kr/]) |
| [[Image:Sum of cubes.png|thumb|right|300px|Visual demonstration that the square of a triangular number equals a sum of cubes.]]
| |
| | |
| In [[number theory]], the sum of the first ''n'' [[Cube (algebra)|cube]]s is the [[Square number|square]] of the ''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'''.
| |
| | |
| ==History==
| |
| 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". {{harvtxt|Pengelley|2002}} finds references to the identity not only in the works of [[Nicomachus]] in what is now [[Jordan]] in the first century CE, but also in those of [[Aryabhata]] in [[India]] in the fifth century, and in those of [[Al-Karaji]] circa 1000 in [[Persia]]. {{harvtxt|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 (number)|0]], [[1 (number)|1]], [[9 (number)|9]], [[36 (number)|36]], [[100 (number)|100]], 225, 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 ''n''×''n'' [[Square lattice|grid]]. For instance, the points of a 4×4 grid (or a square made up of 3 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.
| |
| | |
| 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> be not less than any other is equal to the probability that both <math>Y</math> be not less than <math>X</math> and <math>W</math> be not less than <math>Z,</math> that is, <math>\scriptstyle \mathbb{P}\left(\{\max(X,Y,Z)\leq W\}\right)=\mathbb{P}\left(\{X\leq Y\} \cap \{Z\leq W\}\right).</math> Indeed, these probabilities are respectively the left and right sides of the Nichomacus identity, normalized over <math>n^4.</math>
| |
| | |
| ==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:
| |
| | |
| :<math>
| |
| \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>
| |
| | |
| 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.
| |
| | |
| [[File:Nicomachus theorem.svg|thumb |240px |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}}.]]
| |
| In the more recent mathematical literature, {{harvtxt|Stein|1971}} uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also {{harvnb|Benjamin|Quinn|Wurtz|2006}}); he observes that it may also be proved easily (but uninformatively) by induction, and states that {{harvtxt|Toeplitz|1963}} provides "an interesting old Arabic proof". {{harvtxt|Kanim|2004}} provides a purely visual proof, {{harvtxt|Benjamin|Orrison|2002}} provide two additional proofs, and {{harvtxt|Nelsen|1993}} gives seven geometric proofs.
| |
| | |
| ==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.
| |
| | |
| {{harvtxt|Stroeker|1995}} studies more general conditions under which the sum of a consecutive sequence of cubes forms a square. {{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.
| |
| | |
| == References ==
| |
| {{refbegin|colwidth=30em}}
| |
| *{{citation
| |
| | last1 = Benjamin | first1 = Arthur T. | author1-link = Arthur T. Benjamin | last2 = Orrison | first2 = M. E.
| |
| | title = Two quick combinatorial proofs of <math>\scriptstyle \sum k^3 = {n+1\choose 2}^2</math>
| |
| | journal = [[College Mathematics Journal]]
| |
| | year = 2002
| |
| | volume = 33
| |
| | issue = 5
| |
| | pages = 406–408
| |
| | url = http://www.math.hmc.edu/~orrison/research/papers/two_quick.pdf}}.
| |
| *{{citation
| |
| | doi = 10.2307/27646391
| |
| | last1 = Benjamin | first1 = Arthur T. | author1-link = Arthur T. Benjamin | last2 = Quinn | first2 = Jennifer L. | last3 = Wurtz | first3 = Calyssa
| |
| | title = Summing cubes by counting rectangles
| |
| | journal = [[College Mathematics Journal]]
| |
| | year = 2006
| |
| | volume = 37
| |
| | issue = 5
| |
| | pages = 387–389
| |
| | url = http://www.math.hmc.edu/~benjamin/papers/rectangles.pdf
| |
| | jstor = 27646391}}.
| |
| *{{citation
| |
| | last = Bressoud | first = David
| |
| | authorlink = David Bressoud
| |
| | title = Calculus before Newton and Leibniz, Part III
| |
| | url = http://www.macalester.edu/~bressoud/pub/CBN3.pdf
| |
| | publisher = AP Central
| |
| | year = 2004}}.
| |
| *{{citation
| |
| | last1 = Garrett | first1 = Kristina C. | last2 = Hummel | first2 = Kristen
| |
| | title = A combinatorial proof of the sum of ''q''-cubes
| |
| | journal = [[Electronic Journal of Combinatorics]]
| |
| | volume = 11
| |
| | year = 2004
| |
| | issue = 1
| |
| | at = Research Paper 9
| |
| | mr = 2034423
| |
| | url = http://www.combinatorics.org/Volume_11/Abstracts/v11i1r9.html}}.
| |
| *{{citation
| |
| | last1 = Gulley | first1 = Ned
| |
| | editor-last = Shure | editor-first = Loren
| |
| | title = Nicomachus's Theorem
| |
| | url = http://blogs.mathworks.com/loren/2010/03/04/nichomachuss-theorem/
| |
| | date = March 4, 2010
| |
| | publisher = Matlab Central}}.
| |
| *{{citation
| |
| | doi = 10.2307/3219288
| |
| | last = Kanim | first = Katherine
| |
| | title = Proofs without words: The sum of cubes—An extension of Archimedes' sum of squares
| |
| | jstor = 3219288
| |
| | journal = [[Mathematics Magazine]]
| |
| | volume = 77
| |
| | issue = 4
| |
| | year = 2004
| |
| | pages = 298–299}}.
| |
| *{{citation
| |
| | last = Nelsen | first = Roger B.
| |
| | title = Proofs without Words
| |
| | publisher = Cambridge University Press
| |
| | year = 1993
| |
| | isbn = 978-0-88385-700-7}}.
| |
| *{{citation
| |
| | last = Pengelley | first = David
| |
| | contribution = The bridge between continuous and discrete via original sources
| |
| | url = http://www.math.nmsu.edu/~davidp/bridge.pdf
| |
| | title = Study the Masters: The Abel-Fauvel Conference
| |
| | year = 2002
| |
| | publisher = National Center for Mathematics Education, Univ. of Gothenburg, Sweden}}.
| |
| *{{citation
| |
| | doi = 10.2307/2688231
| |
| | last = Stein | first = Robert G.
| |
| | title = A combinatorial proof that <math>\scriptstyle \sum k^3 = (\sum k)^2</math>
| |
| | journal = [[Mathematics Magazine]]
| |
| | volume = 44
| |
| | issue = 3
| |
| | pages = 161–162
| |
| | year = 1971
| |
| | jstor = 2688231}}.
| |
| *{{citation
| |
| | last = Stroeker | first = R. J.
| |
| | title = On the sum of consecutive cubes being a perfect square
| |
| | journal = [[Compositio Mathematica]]
| |
| | volume = 97
| |
| | year = 1995
| |
| | issue = 1–2
| |
| | pages = 295–307
| |
| | mr = 1355130
| |
| | url = http://www.numdam.org/item?id=CM_1995__97_1-2_295_0}}.
| |
| *{{citation
| |
| | title = The Calculus, a Genetic Approach
| |
| | last = Toeplitz | first = Otto
| |
| | authorlink = Otto Toeplitz
| |
| | publisher = University of Chicago Press
| |
| | year = 1963
| |
| | isbn = 978-0-226-80667-9}}.
| |
| *{{citation
| |
| | last = Warnaar | first = S. Ole
| |
| | title = On the ''q''-analogue of the sum of cubes
| |
| | journal = [[Electronic Journal of Combinatorics]]
| |
| | volume = 11
| |
| | year = 2004
| |
| | issue = 1
| |
| | at = Note 13
| |
| | mr = 2114194
| |
| | url = http://www.combinatorics.org/Volume_11/Abstracts/v11i1n13.html}}.
| |
| *{{citation
| |
| | last = Wheatstone | first = C.
| |
| | authorlink = Charles Wheatstone
| |
| | title = On the formation of powers from arithmetical progressions
| |
| | journal = [[Proceedings of the Royal Society of London]]
| |
| | volume = 7
| |
| | pages = 145–151
| |
| | year = 1854
| |
| | doi = 10.1098/rspl.1854.0036}}.
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| *{{mathworld|urlname=NicomachussTheorem|title=Nicomachus's Theorem}}
| |
| *[http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32 A visual proof of Nicomachus's Theorem]
| |
| | |
| {{Classes of natural numbers}}
| |
| | |
| [[Category:Elementary mathematics]]
| |
| [[Category:Number theory]]
| |
| [[Category:Integer sequences]]
| |
| [[Category:Mathematical identities]]
| |
| [[Category:Articles containing proofs]]
| |