Euler's Pentagonal Number Theorem Author(s): George E. Andrews Reviewed work(s): Source: Mathematics Magazine, Vol. 56, No. 5 (Nov., 1983), pp. 279-284 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2690367 . Accessed: 20/02/2013 14:16

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This content downloaded on Wed, 20 Feb 2013 14:16:29 PM All use subject to JSTOR Terms and Conditions Euler's Pentagonal NumberTheorem

GEORGE E. ANDREWS The PennsylvaniaState University UniversityPark, PA 16802

One of Euler's most profounddiscoveries, the PentagonalNumber Theorem [7], has been beautifullydescribed by AndreWeil: Playingwith series and products,he discovereda numberof factswhich to him looked quite isolatedand verysurprising. He looked at thisinfinite product

- - (I x)( X)(-X3). and just formallystarted expanding it. He had manyproducts and seriesof thatkind; in some cases he got somethingwhich showed a definitelaw, and in othercases thingsseemed to be ratherrandom. But with this one, he was verysuccessful. He calculated at least fifteenor twentyterms; the formulabegins like this:

1I(1 -Xn)= 1 -x-X2+ X5 + x7-x12 X15

where the law, to your untrainedeyes, may not be immediatelyapparent at firstsight. In modem notation,it is as follows:

00 + 00

I(I-q n) = E: (_ I)n qn(3n+ 1)/2(1 1 -00

whereI've changedx intoq sinceq has become thestandard notation in ellipticfunction-theory since Jacobi. The exponentsmake up a progressionof a simple nature. This became im- mediatelyapparent to Euler afterwriting down some 20 terms;quite possiblyhe calculated about a hundred.He veryreasonably says, "this is quite certain,although I cannotprove it;" ten yearslater he does proveit. He could not possiblyguess thatboth seriesand productare partof thetheory of ellipticmodular functions. It is anothertie-up between number-theory and ellipticfunctions [22, pp. 97-98]. G. P6lya [16, pp. 91-98] providesa moreextensive account of Euler'swonderful discovery togetherwith a translationof Euler'sown description[6]. The numbersn(3n - 1)/2 are called "pentagonalnumbers" because of theirrelationship to pentagonalarrays of points.FIGURE 1 illustratesthis. Legendre [14, pp. 131-133]observed that purelyformal multiplication of theterms on theleft side of (1) producesthe term + qN precisely as oftenas N is representableas a sumof distinctpositive integers; the " + " is takenwhen there is an evennumber of summandsin therepresentation and the"-" whenthe number of summands is odd. For example,

n=1 n=2 n=3 n=4

FIGURE 1. The firstfour pentagonal numbers are 1, 5, 12, 22.

VOL. 56, NO. 5, NOVEMVBER1983 279

This content downloaded on Wed, 20 Feb 2013 14:16:29 PM All use subject to JSTOR Terms and Conditions (1 -q)(I -q2)(1 - q3)(1 - q4) ... Ilqq2 q3 q4+q1+2+q1+3+ q1+4 +q2+3

+q2+4 + q3+4 1q +2+3 1q1+2+4 1q+3+4 _ 2+3+4 1+ q1+2+3+4 + The termpartition is usuallyused to describethe representation of a positiveinteger as thesum of positiveintegers. In thisarticle, we are concernedwith unordered partitions; two such partitions are consideredthe same if theterms in thesum are thesame, e.g., 1 + 2 and 2 + 1 are considered as the same partitionof 3. Thus Legendre'sobservation may be matchedup withthe actual infiniteseries expansion (1) as follows. THEOREM.Let pe(n) denotethe numberof partitionsof n intoan even numberof distinct summands.Let po(n) denotethe number of partitions of n intoan oddnumber of distinct summands. Then

ifn=j(3j?1)12 p, e(n)n-p(n)= -po(n > ((-W (2) otherwise.

The impactof Euler'sPentagonal Number Theorem and Legendre'sobservations on subse- quentdevelopments in numbertheory is enormous.Both (1) and (2) arejustly famous. Indeed, F. Franklin'spurely arithmetic proof of (2) [10](see also [21,pp. 261-263])has beendescribed by H. Rademacheras thefirst significant achievement of Americanmathematics. Franklin's proof is so elementaryand lovelythat it has been presentedmany times over in elementaryalgebra and numbertheory texts [5, pp. 563-564],[11], [12, pp. 206-207],[13, pp. 286-287],[15, pp. 221-222]. It is, however,interesting to notethat Euler's proof of (1) alludedto by Weil remainsalmost unknown.In recentyears only Rademacher's book has containedan expositionof it [17, pp. 224-226].This book and earlierbooks [4, pp. 23-24] havepresented it moreor less as Eulerdid. Whilethe idea behindEuler's proof is ingenious(as one wouldexpect), the mathematical notation of Euler'sday hidesthe fact that other results of significanceare eithertransparent corollaries of Euler'sproof or lie just below the surface.The remainderof thisarticle is devotedto a long overduemodem exposition of Euler'sproof and an examinationof its consequences. To begin,we definea functionof twovariables:

00 .. I f( , q) E (Il xq)(I - xq 2) **(-xqt- )X n+1qn.3 n = I

(Absoluteconvergence of all seriesand productsconsidered is ensuredby Iq I< 1, Ix j lq1)q We firstnote that

00 f(I, q) =H ( - qt"). (4) n==I

This is becausewe mayeasily establish the identity

N N ... I - E (I - q)(I - q2) (I - qt- 1) qtl H (I - qn) (5 n= I II= I by mathematicalinduction on N (a niceexercise for the reader). Thus (4) is thelimiting case of(5) as N-> oo. The main step in Euler's proof is essentiallythe verificationof the followingfunctional equation:

f (x q)= 1 - x2q - x3q2f(xq q). (6)

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This content downloaded on Wed, 20 Feb 2013 14:16:29 PM All use subject to JSTOR Terms and Conditions ActuallyEuler does equation(6) overand over,first with x = 1,then x = q, thenx = q2, and so on [7,pp. 473-475];page 473 ofhis paper in OperaOmnia is shownon thenext page. This repetition of specialcases of (6) tendsto hidewhat exactly is happening.To prove(6), we takethe defining equation(3) througha sequenceof algebraicmanipulations:

00 ... n f(xgq I_X2q- E (I -Xq)(I1 -xq 2) (I -xq n- ) Xn+l q n=2 00

= -x2q- E (1 -xq)(I -xq2) *- (I -xqn)Xn+2qn+ n = 1 00 ... (I - = x 2q- E (I - xq2) -xq n) xn+2q II+I(I xq) n = 1

oo = - 2 2 n+2 n+1 q- E (I -xq- ) ... (I -xqn)- X 2q n=1

oo

+ E (1 -xq2) *.. (I -xqn)xn+3qn+2 n = 1 00 - ... n+2 I+ I = -x 2q-x 3q 2- , (I xq2) (I - xq )x q n=2

oo ... + E (I -xq2) (I xq")Xn+3qn+2 n = I 00 _ I -x2q-x3q2- (I -xq 2)- * (I -xqn )X q n= 1

... + E (1 -xq2) (I xqn)xn+3qn+2 n = 1 00 2 ... I x 2q -x 3q E (I -xq 2)- (I -xqn)Xn+3 qn+2((l _xq n+l 1) n = 1

2 =I x 2q-x 3q { E (I -xq 2) .(1 xq n)x,,1 q2n)?I n = 1

= 1-x2q-x3q2f (xq, q)-

If youfollowed the above sequence of stepscarefully, you see howat each stagethings seem to fit togethermagically at just the rightmoment. Also you can appreciatethe complicationof repeatedpresentation of thesame steps first with x = 1,then x = q, thenx = q2, etc.However, the empiricaldiscovery of (6) by Eulermust have comeprecisely in thisrepetitive manner. The restof Euler'sproof is now almostmechanical. Equation (6) is repeatedlyiterated; thus

f (x, q)= 1X2q-X3q2(1 X2q3_X3q5f (xq2 q))

=l1-x2q-x3q2 + x5q5 + x6q7(l_X2q5_x3q8f (xq3, q))

(7) N-i

= 1 + n(3n+1)/2)(_ I)n(X3n-lqn(3n-1)/2 + X31q n= 1

+ (_ 1)Nx3N-lqN(3N-1)/2 + (- )Nx3NqN(3N+ 1)/2f ( xqN, q)

VOL. 56, NO. 5, NOVEMBER 1983 281

This content downloaded on Wed, 20 Feb 2013 14:16:29 PM All use subject to JSTOR Terms and Conditions 47-48] EVOLUTIO PRODUCTI INFINITI (I -x)(l -xx)(I -x')(l -zx) etc. 473

2. Quoniam hi termini omnes factoremhabent communem1 - x, eo evoluto singuli terminidiscerpentur in binas partes, quas ita repraesentemus

A - xx + x(l - xx) + x'(1 - xx)(1 - x) + x'(1 - xx)(1 - x)(1 - x') + etc. -x-.x' (1-zx)-x5(1-xx)(1-x)-x`(1-xx)(1-x')(1-x')-etc. Hinc iam binae partes eadem potestate ipsius x affectaein unam contrahan- tur ac resultabitpro A sequens forma

------A -xx x6- x(1 xx) -x(1-xx$) (1-x) x"(1 x)(1 x(l ') -etc.,

ubi duo terminiprimi xx - x5 iam sunt evoluti; sequentes autem procedunt per has potestatesx7, x9, x11,x S, x15, quarum exponentesbinario crescunt.

3. Ponamus nune simili modo ut ante

A{= xx - x5- B, ita ut sit

B = x7(1-xx)+x1 1 - x) (1-x3) + xl(l -xx) (t -x) (1 - Z') + etc., cuius omnes terminihabent factoremcommunem 1 - xx, quo evoluto singuli terminiin binas partes discerpantur,uti sequitur,

B =7 + x9(1 - x8) + xll(l - x3)(1-- x') + x13(1-x3 (1- x4)(1 - x) + etc.

X ( - - ll(l- x)3( - x) - 1l(1 -xl3)(1- 4) (1 - x5) - etc. Hic iterum bini termini,qui eandem.potestatem ipsius x habent praefixam, in unam colliganturet prodibit

B = x7- x1- x5(1 - x) - x18(1- x3)(1 - x4)-x2(1 -x)(1 -z4)(1 -x)-etc.

ubi iani potestatesipsius x crescuntternario.

4. Statuaturnunc porro B= x7- xs - C, ita ut sit X C-2']5( -- x5) + v,18(1- Xt)(1 - ,4) + j21( $S) ( X-Ix) (1-I5) + etc., LRONnqitmEut.um Opera oinnia I3 (Commentationesarithucticae 60

anotherfine exercise in mathematicalinduction on N. In thelimit as N -> oo we find 00 n f(x, q)= 1 + E (-1) (X3n- lqn(3n-1)/2 + X3nqn(3t+ 1)/2) (8) n= 1 Thereforeby (4) and (8), 00 00 n H (I - qt') =f (I, q) = + ,: (_ I) (qn(3n -1)/2 + qnl(3n+ 1)/2) n=l n=l

00 = (_ l) qn(3n+ 1)/2, n= -oo whichcompletes the proof of (1). It shouldbe notedthat several authors (L. J.Rogers [18, pp. 334-335],G. W. Starcher[19], and N. J.Fine [9]) also foundformula (8) essentiallyin theway we have;however, none has notedthat he was, in fact,rediscovering Euler's proof in simplerclothing. M. V. Subbarao[20] has also shownthe connection between (8) and Franklin'sarithmetic proof [10].

282 MATHEMATICS MAGAZINE

This content downloaded on Wed, 20 Feb 2013 14:16:29 PM All use subject to JSTOR Terms and Conditions Now thereader may naturally ask: Was anythinggained in thisgeneral formulation of Euler's proofbesides simplicity of presentation? Is anynew information available that was hiddenbefore? We can answera resoundingYES just by settingx = -1 in (3) and (8). Thissubstitution gives the equation

00 1 + E (1 +q)(1 +q2) (1 + qn1)(- l)nqn

n= 1 (9) = f -1,q) =1 +E (q _n(3n- 1)/2+ qn(3n+ 1)/2) n = I Equation(9) yieldsa corollaryas appealingas Legendre'sTheorem. It impliesthat the product

(I + q)(I + q 2) .. (I + q n- I)qn whenmultiplied out produces the termqN exactlyas oftenas N can be partitionedinto distinct summandswith largest part equal to n. For example,when n = 4, (1 + q)(I + q2)(1 + q3)q4 = q4+ q4+1 + q4+2 + q43 + q4+2+1

+ 4+3+1 + q4+3+2 + q4+3+2+?1 Hence theseries on theleft side of (9) whenexpanded out yieldsthe term ? qN foreach partition of N intodistinct summands; the " + " occursif thelargest summand is even,and the" - " occurs if thelargest summand is odd. In thesame manner that (1) yieldedLegendre's Theorem, we see that(9) yieldsthe followingequally elegant but littlepublicized result found by N. J. Fine [8] morethan 118 years after Legendre's observation. THEOREM. Let 7r,(n)denote the number of partitions of n withdistinct summands the largest of whichis even.Let 7r0(n)denote the number of partitions of n withdistinct summands the largest of whichis odd. Then { 1 ifn=j(3j+ 1)/2, qre(n)-gT (n)= -1 ifn =j(3j- 1)/2, O otherwise. Let us checkout thetheorems with an example.The partitionsof n = 12 intodistinct parts are: 12, 11+1, 10+2, 9+3, 9+2+1, 8+4, 8+3+1, 7+5, 7+4+1, 7+3+2, 6+5+1, 6 + 4 + 2, 6 + 3 + 2 + 1, 5 + 4 + 3, 5 + 4 + 2 + 1. The partitionsenumerated by each of p,(l2), p0(12), g7,(12)and g7r(12)are listedin thefollowing table:

P,1_2) ( po(12) qre( 12) qro( 12) 11+1 12 12 11+1 10+2 9+2+ 1 10+2 9+3 9+3 8+3+1 8+4 9+2+1 8+4 7+4+1 8+3+1 7+5 7+5 7+3+2 6+5+1 7+4+1 6+3+2+1 6+5+1 6+4+2 7+3+2 5+4+2+1 6+4+2 6+3+2+1 5+4+3 5+4+3 5+4+2+1.

- - Thus pe(12) -p0(12) = 7 - 8 = - 1 and 7r,(12) 7r.(12)= 7 8-- 1, as predicted by our theo- rems. Beyondthis immediate pleasant discovery that Euler's approach, properly modernized, yields Fine's Theorem,we mayask: Are thereinteresting extensions of Euler'smethod that yield more thanequation (8)? Here againthe answer is positive.L. J.Rogers [18, p. 334],apparently unaware

VOL. 56, NO. 5, NOVEMBER1983 283

This content downloaded on Wed, 20 Feb 2013 14:16:29 PM All use subject to JSTOR Terms and Conditions of howEuler's proof worked, showed in almostprecisely the way we proved(8) that 1 (l - a)(l - aq). (1. - aqn-I)tt 1 -at ... I n=lI (I -b)(I -bq) (1- bqn- 1t

- .0(I- a)(I - a) ..(I - an I) _atq( I_ ...2 (Il- _ )ntnqn2n(I _ atq2n)

- .. - .. n+ n=~ ~ ~~( (1b)(I -bq) (I(1bq )( )- tq) (I(-tq I) (10) Rogersin factshowed that if f(a, b, t) denotesthe left side of (10), then 1 -at (I-a)b- t)(11 f(a, b t)= I_ + t- ))( ))(aq,bq, tq).(1 t (Il-b)(l -t) This resultand deeperextensions of it thatrequire much more than Euler's method have had a majorimpact in thetheory of partitions[1, Secs. 3 and 4], [2, Ch. 7], [3, Ch. 3]. N. J.Fine [9] also independentlyrediscovered (10). Surelythe storyunfolded here emphasizeshow valuableit is to studyand understandthe centralideas behind major pieces of mathematics produced by giants like Euler. The discoveriesof theoremsas appealingas the two we have describedwould not be separatedby 118 yearsif studentsof additivenumber theory had followedthis advice. Thispaper was partially supported by National Science Foundation Grant MCS-8201733.

References [I] G. E. Andrews,Two theorems of Gauss and allied identities proved arithmetically, Pacific J. Math., 41 (1972) 563-578. [21 , The Theoryof Partitions,vol. 2, Encyclopediaof Mathematicsand Its Applications,Addison- Wesley,Reading, 1976. [3] , Partitions:Yesterday and Today, New Zealand Math. Soc., Wellington, 1979. [4] P. Bachmann,Zahlentheorie: Zweiter Teil, Die AnalytischeZahlentheorie, Teubner, Berlin, 1921. [5] G. Chrystal,Algebra, Part II, 2nded., Black, London, 1931. [6] L. Euler,Opera Omnia, (1) 2, 241-253. [7] , Evolutioproducti infiniti (I - x)(I - xx)(I - x3)(1 - x4)(1 - x5)(1 - X6) etc.,Opera Omnia, (1) 3, 472-479. [8] N. J.Fine, Some new results on partitions,Proc. Nat. Acad. Sci. U.S.A.,34 (1948)616-618. [9] , SomeBasic Hypergeometric Series and Applications, unpublished monograph. [10] F. Franklin,Sur le developpementdu produit infini (I - x)(I - x2)(1 - X3) * -*, ComptesRendus, 82 (1881) 448-450. [11] E. Grosswald,Topics in theTheory of Numbers, Macmillan, New York, 1966; Birkhiuser, Boston, 1983. [12] H. Gupta,Selected Topics in NumberTheory, Abacus Press, Tunbridge Wells, 1980. [13] G. H. Hardyand E. M. Wright,An Introductionto the Theory of Numbers, 4th ed., Oxford University Press, London,1960. [14] A. M. Legendre,Theorie des Nombres,vol. II, 3rd.ed., 1830(Reprinted: Blanchard, Paris, 1955). [15] I. Nivenand H. Zuckerman,An Introductionto theTheory of Numbers,3rd ed., Wiley, New York, 1972. [16] G. Polya,Mathematics and PlausibleReasoning, Induction and Analogy in Mathematics,vol. 1,Princeton, 1954. [17] H. Rademacher,Topics in Analytic Number Theory, Grundlehren series, vol. 169, Springer, New York, 1973. [18] L. J.Rogers, On twotheorems of combinatory analysis and some allied identities, Proc. London Math. Soc., (2) 16 (1916)315-336. [19] G. W. Starcher,On identitiesarising from solutions of q-differenceequations and someinterpretations in numbertheory, Amer. J. Math., 53 (1930)801-816. [20] M. V. Subbarao,Combinatorial proofs of someidentities, Proc. Washington State University Conf. on NumberTh',ory, Pullman, 1971, 80-91. [21] J.J. Sylvester, A constructive theory of partitions, arranged in threeacts, an interactand an exodion,Amer. J.Math., 5 (1882)251-330. [22] A. Weil,Two lectures on numbertheory, past and present, L'Enseignement Mathematique, 20 (1974)87- 110.

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