IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 4 (Jul. - Aug. 2013), PP 21-22 www.iosrjournals.org www.iosrjournals.org 21 | Page Pythagorean Triangle and Special Pyramidal Numbers M. A. Gopalan 1 , V. Sangeetha 2 , Manju Somanath 3 1 Department of Mathematics,Srimathi Indira Gandhi College,Trichy-2,India 2,3 Department of Mathematics,National College, Trichy-1,India Abstract: Patterns of Pythagorean triangle, where, in each of which either a leg or the hypotenuse is a pentagonal pyramidal number and Centered hexagonal pyramidal number, in turn are presented. Keywords: Pythagorean triangles, pentagonal pyramidal,centered hexagonal pyramidal. I Introduction The method of obtaining three non-zero integers ,  and  under certain relations satisfying the equation  2 +  2 =  2 has been a matter of interest to various mathematicians [1,2,3].In [4-12], special Pythagorean problems are studied.In this communication, we present yet another interesting Pythagorean problem.That is, we search for patterns of Pythagorean triangles where in each of which, either a leg or the hypotenuse is represented by a pentagonal pyramidal number and centered hexagonal pyramidal number,in turn. II Notation    - m-gonal pyramidal number of rank n    - centered m-gonal pyramidal number of rank n   , - polygonal number of rank n. III Method of Analysis Let (m,n,k) represent a triple of non-zero distinct positive integers such that  =( + 1) Let , ,  be the Pythagorean triangle whose generators are m,n. Consider  =2 ;  =  2 − 2 ;  =  2 +  2 . It is observed that, for suitable choices of n, either a leg or hypotenuse of the Pythagorean triangle P is represented by a pentagonal pyramidal number and centered hexagonal pyramidal number ,in turn.Different choices of n along with the corresponding sides of the Pythagorean triangle are illustrated below Choice 3.1 Let  =4 +3. The corresponding sides of the Pythagorean triangle are  = 32 3 + 80 2 + 66 + 18  = 16 4 + 56 3 + 57 2 + 18  = 16 4 + 56 3 + 89 2 + 66 + 18 Note that  =   5 Choice 3.2 Let  =2 2 +4 +3 The corresponding sides of the Pythagorean triangle are  =8 5 + 40 4 + 88 3 + 104 2 + 66 + 18  =4 6 + 24 5 + 60 4 + 80 3 + 57 2 + 18  =4 6 + 24 5 + 68 4 + 112 3 + 113 2 + 66 + 18 Note that  =   5 Note It is worth mentioning here that, for the following two choices of m,n given by (i)  =4,  =  +1 and (ii)  =2 3 − 3,  =  the sides  and  represent   5 respectively. Choice 3.3 Let  = 2( + 1) The corresponding sides of the Pythagorean triangle are  =8 3 + 24 2 + 24 +8  =4 4 + 16 3 + 20 2 +8  =4 4 + 16 3 + 28 2 + 24 +8 Note that  =   6