GENERAL ¨ ARTICLE 'Triangular Numbers ' Anuradha S Garge and Shailesh A Shirali The triangularnumbers, which are numbers as- sociatedwithcertain arrays of dots, were known to the ancient Greeks and viewed by them with reverence. Though possessing a simple de¯ni- tion, they are astonishingly rich in properties of Left: Anuradha S Garge various kinds, ranging from simple relationships completed her PhD from between them and the square numbers to very Pune University in 2008 under the supervision of Prof. complex relationships involving partitions, mod- S A Katre. Her research ular forms, etc. { topics which belong to advanc- interests include K-theory ed mathematics. They also possess many pretty and number theory.Besides combinatorialproperties. In this expository ar- mathematics, she is interested ticlewesurveya few of these properties. in (singing) indian classical music and yoga. Triangularnumbers are numbers associated with trian- Right: Shailesh Shirali is gular arrays of dots. The idea is easier to convey using Director of Sahyadri School (KFI), Pune, and also Head of pictures rather thanwords;seeFigure 1. We see from the Community Mathematics the ¯gure thatif Tn denotes the n-th triangularnumber, Centre in Rishi Valley School then T1 =1,T2 = T1 +2,T3 = T2 +3,T4 = T3 +4,.... (AP).Hehasbeeninthefield Thus Tn = Tn 1 + n,leadingto: of mathematics education for ¡ three decades, and has been n(n +1) closely involved with the Tn =1+2 + +(n 1) + n = : (1) Math Olympiad movement in ¢¢¢ ¡ 2 India. He is the author of The triangular numbers appearinthePascal's triangle many mathematics books addressed to highschool along the third diagonal; in Figure 2they are the num- students, and serves as an bers in bold type. editor for Resonance and for At Right Angles.Heis engaged in many outreach Figure 1. The first four triangular numbers. projects in teacher education. Keywords Triangular number, figurate num- ber, rangoli, Brahmagupta–Pell equation, Jacobi triple product T1=1 T2=3 T3=6 T4=10 identity. 672 RESONANCE ¨July 2012 GENERAL ¨ ARTICLE Figure 2. Pascal’s triangle. Triangular numbers were known to the ancient Greeks Triangular numbers and were viewed by them with mysticalreverence. The are numbers triangularnumber10wasconsideredtobe a symbol of associated with `perfection', being the sum of 1(a point), 2(a line),3 triangular arrays of (a plane) and 4 (a solid). dots. For a sequence de¯ned in such a simple manner, the se- quence of triangularnumbersabbreviated as T-numbers is astonishingly rich in properties. We explore many of them in this article. 1. Some Properties of TriangularNumbers 1.1 Triangular Numbers Moduloan Integer Here we look at the triangularnumbersmoduloa posi- tive integer k.Reading the triangularnumbers modulo 2, we get the following pattern which repeats every 4 steps: 1; 1; 0; 0; 1; 1; 0; 0; 1; 1; 0; 0; :::: Similarly, reading the numbers modulo 3 gives the fol- lowing pattern of numbers which repeats every 3 steps: 1; 0; 0; 1; 0; 0; 1; 0; 0; 1; 0; 0; :::; Triangular numbers modulo 4 we get a sequence thatrepeats every 8 steps: were known to the ancient Greeks and 1; 3; 2; 2; 3; 1; 0; 0; 1; 3; 2; 2; 3; 1; 0; 0; :::; were viewed by them with mystical modulo 5 we get a sequence thatrepeats every 5 steps: reverence. RESONANCE ¨ July 2012 673 GENERAL ¨ ARTICLE 1; 3; 1; 0; 0; 1; 3; 1; 0; 0; :::; and modulo 6, we get this pattern: 1; 3; 0; 4; 3; 3; 4; 0; 3; 1; 0; 0; 1; 3; 0; 4; :::: After some experimentation,weare able to guess that 52 = T +T 4 5 the T-residues mod k repeateveryk steps if k is odd, and every 2k stepsif k is even (see [1]). Once noticed, it is rather easy toprove. 1.2 Triangular Numbers and Squares The Greeks knew thatthesum of a pair of consecutive triangularnumbers is a perfect square. The proof of this 62 = T +T using algebra is trivial, but Figure 3shows a nice `proof 5 6 without words' which clearly generalizes. The relation 2 Figure 3. Pictorial patterns in Figure 4(8T3 +1 = 49 = 7 ) also generalizes in an explaining why the sum of obvious way; it too wasknown to the Greeks. Proba- two consecutive T-numbers bly hidden inside some well-known `rangoli patterns' are is a perfect square. more such identities for triangularnumbers. 1.3 Some Identities The following combinatorialinterpretation maybegiven to the T-numbers: Tn is the number of ordered pairs (x; y), where 1 dddx y n; x; y N. Using this we may provesomeidentities. For example,2 the identity Figure 4. A ‘rangoli’ pattern T = T + T + mn; (2) with 4-fold rotational sym- m+n m n metry; it explains why eight is proved as follows: counts the number of pairs times a T-number plus 1 is a Tm+n perfect square. (x; y)with1dddxym+n.Wepartitionthecollection of all such pairs into three classes: (i) pairs (x; y)with 1dddxym;(ii)pairs (x; y) with m+ 1 dddxym+n; and (iii) pairs (x; y)with1dddxm<ym+ n.Count- ing these three classes separately, we get (2). Relation (2) has the following unexpected consequence. Let a heapof beads (any number) be placed on a table. It is divided it into two heaps in any wayonepleases, and 674 RESONANCE ¨July 2012 GENERAL ¨ ARTICLE the product of the sizes of the two new heapsisrecorded. There exist many This step is then iterated: each of the resulting heaps connections within is subdivided into two new heaps, etc. After some steps the sequence of T- each heapwillhave just one bead and no further divi- numbers, and sion is possible. Now let all the products recorded be between the T- summed. It will be found that whichever wayonedoes numbers and the it, the totalobtained in the end is the same; it depends squares. only on the number of beads in the starting heap. Try it out, and explain how it follows from (2). Another such relation is: Tmn = TmTn + Tm 1Tn 1;(3) ¡ ¡ it hasthefollowing corollary, which is obtained by putting m = n: 2 2 Tn2 =(Tn) +(Tn 1) : (4) ¡ Thus, there exist many connections within the sequence of T-numbers, and between the T -numbers and the squa- res. For more such examples see [2]. These properties, once spotted, are easy to prove, but the greater chal- lenge lies in ¯nding them. 1.4 Relations Among the Triangular Numbers Here are some more relations among the triangularnum- bers which are easy to verify algebraically: If n is a triangular number, then so is 9n +1. (5) If n is a triangular number, then so is 25n + 3. (6) 1 For example, for (5): if n = 2 k(k +1),then 9n + 1 = 9 1 2 k(k +1)+1 = 2 (3k + 1)(3k +2). Eulerposed (and answered) the following problem: Find all integer pairs (r; s) which satisfy the following condition: If n is a triangular number, then so is rn+s. Going deeper, we get the following result whose proof we leave as anexercise: RESONANCE ¨ July 2012 675 GENERAL ¨ ARTICLE It is of interest to Theorem 1.1. The integer pairs (r; s) for which the know which property`n triangular implies rn + s triangular' holds triangular numbers are those for which r is an odd square, and s = 1 (r 1). 8 ¡ are themselves Fittingly, each such s is a triangularnumber. For more squares. such relations, see[3]. 2. Which TriangularNumbersare Squares? Having seen some connections between triangularnum- bers and squares, it is of interest to know which tri- angularnumbers are themselves squares. This question See Resonance issues on Euler, Gauss, Brahmagupta: has been well-studied, and one hasthefollowing striking Vol.2, Nos 5 & 6, 1997; result originally provedbyEulerin1730. Vol.17, No.3, 2012. Theorem 2.1. A triangular number Tn is a square if andonly if n has the form 2k 2k p2 + 1 + p2 1 2 n = ¡ ¡ ; 4 ¡ ¢ ¡ ¢ where k is an integer. In particular, there exist in¯nitely many triangular numbers which are squares. 1 2 Aquickproof goes as follows: if 2 n(n+1)=m for some m,then(2n +1)2 8m2 =1.Bya suitable change of variables (x =2n+1,¡ y =2n), we get theBrahmagupta{ Pell equation which hasboth a rich history and a rich theory behind it: x2 2y2 =1: ¡ Solving this we get the ¯rst few values of n as1,8,49 and 288, with corresponding T -values 1, 36, 1225 = 352; and 41616 = 2042. There are many ways of arriving at the generalsolution of the Brahmagupta{Pell equation. One approach is to The Brahmagupta– consider the number ¯eld Q(p2) and to de¯ne a function Pell equation has f on this ¯eld as follows: if x = a + bp2 where a; b Q, both a rich history 2 then f(x)=a2 2b2. We seek all elements of Z(p2) with and a rich theory unit value for f¡.Itis easy to check that f is multiplica- behind it. tive; i.e., if x; y Q(p2), then f(xy) = f(x)f(y). Let 2 676 RESONANCE ¨July 2012 GENERAL ¨ ARTICLE u =1+p2. Since f(u)= 1, it follows that f(u2k)=1 The equation in ¡ for any integer k. Following through with this observa- Section 2.2 is more tion we get the `if'part of the above result. The `only difficult to solve than if'part is obtained by a descent procedure, in which the Brahmagupta– from any given solution we construct a smalleroneby Pell equation; it multiplying by v =1 p2. involveselliptic ¡ 2.2 ...and Cubes? curves. Anaturalquestiontoask now is: do there exist tri- angular numbers that are perfect cubes? If there are such numbers, then one cansimplify as above togetan equation of the form: (2n +1)2 (2m)3 =1: ¡ This equation is more di±cult to solve than the Brahma- gupta{Pell equation;indeed,itinvolvesquestions about elliptic curves.