Fascinating Triangular Numbers

 

Displaying Triangular Numbers

The numbers which can be arranged in a compact triangular pattern are termed as triangular numbers. The triangular numbers are formed by partial sum of the series 1+2+3+4+5+6+7......+n. So

T1 = 1
T2 = 1 + 2 = 3
T3 = 1 + 2 + 3 = 6
T4 = 1 + 2 + 3 + 4 = 10

So the nth triangular number can be obtained as Tn = n*(n+1)/2, where n is any natural number.In other words triangular numbers form the series 1,3,6,10,15,21,28.....

Flocks of birds often fly in this triangular formation. Even several airplanes when flying together constitute this formation. The properties of such numbers were first studied by ancient Greek mathematicians, particularly the Pythagoreans.

Have you heard of the following famous story about the famous mathematician Carl F. Gauss.

" The teacher asked everyone in the class to find the sum of all the numbers from 1 to 100. To everybody's surprise, Gauss stood up with the answer 5050 immediately. The teacher asked him as to how it was done. Gauss explained that instead of adding all the numbers from 1 to 100, add first and last term i.e. 1 + 100 =101, then add second and second last term i.e. 2 + 99 =101 and so on. Every pair sum is 101 and their will be 50 such pairs ( total 100 numbers in all to be added), so 101 * 50 = 5050 is the answer. So the sum of numbers from 1 to N is (N/2)*(N+1), where N/2 are the number of pairs and N+1 is sum of each pair. This the famous formula for nth triangular number."

Some of the interesting properties of triangular numbers published in [5] are:

Curious properties of Triangular Numbers:

T1 + T2 = 1 + 3 = 4 = 22
T2 + T3 = 3 + 6 = 9 = 32

9*T1 + 1 = 9 * 1 + 1 = 10 = T4
  9*T2 + 1 = 9 * 3 + 1 = 28 = T7

8*T1 + 1 = 8 * 1 + 1 = 9 = 32
  8*T2 + 1 = 8 * 3 + 1 = 25 = 52

T42 = 102 = 100 = 13 + 23 + 33 + 43
T52 = 152 = 225 = 13 + 23 + 33 + 43 + 53

T12 + T22 = 1 2 + 3 2= 10 = T4
 T22 + T32 = 3 2 + 6 2= 45 = T9
  T32 + T42 = 6 2 + 10 2= 136 = T16
   
Tn-12 + Tn2 = Tn2
 

(Dr. Diego Marques, University of Brasilia, Brazil submitted vide his email dated 20 Apr 2011 that "this is also valid for the amazing Fibonacci sequence i.e. The sum of the squares of two consecutive fibonacci numbers is also a fibonacci number).

T12 = 1 * 1 = 1 = T1
  T32 = 6 * 6 = 36 = T8

Can anybody find the third triangular number whose square is also a triangular number ?.

(AMRIK S NIMBRAN from Patna, Bihar submitted vide his email dated 21 Dec 2011 that "It took me some time to locate the proof of the impossibility of any other square triangular number beside 1 and 36 which is square of a triangular number, The source of proof is: L. J. Mordel, Diophantine Equations, 1969, Academic Press, London, Theorem 7, pp. 268-269).

21 + 15 = 36 = T8 : 21 - 15 = 6 = T3

171 + 105 = 276 = T23 : 171 - 105 = 66 = T11

703 + 378 = 1081 = T46 : 703 - 378 = 325 = T25

     and so on.

 

Some New Observations on Triangular Numbers :

4 * 5 * 6 = 2 * 3 * 4 * 5 = 1 * 2 * 3 * 4 * 5 = 120

No other triangular number is known to be the product of four or more consecutive numbers.

(submitted by Dr. N Mander (Mr) from Denmark vide his email dated 22 Oct 2011).

(submitted by Mr Roy Blatchford vide his email dated 17 Mar 2011).

Can anybody find similar recursive equations for pairs (a,b) of positive integers such that 3 * Ta=Tb ?.

(submitted by Don Davis vide his email dated 20 June 2013).

(submitted by Okoh Ufuoma Cyrus from Nigeria vide his email dated 26 April 2013).

 

If you find any new and interesting observation about triangular numbers, please email me.

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References:

[1] Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.

[2] Beiler, Albert H. Recreations in the Theory of Numbers. New York: Dover, 1966.

[3] Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.

[4] Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

[5] Gupta, Shyam Sunder "Curious Properties of triangular numbers." Science Reporter , September 1987, India.

[6] Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138 and 147-150, 1994.

[7] Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

[8] Madachy, Joseph S. Madachy's Mathematical Recreations. New York: Dover, 1979.

[9] Pickover, Clifford A. Wonders of Numbers. New York: Oxford University Press, 2001.

[10] Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

[11] Trotter, T. Jr. "Some Identities for the Triangular Numbers." J. Recr. Math. 6, 128-135, 1973. http://www.trottermath.net/numthry/trident.html.

[12] Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1997.

[13] Wells, David. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin Books, 1991.



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This page is created on 26 Oct, 2002.

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