IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 5 (Jul. - Aug. 2013), PP 62-67 www.iosrjournals.org www.iosrjournals.org 62 | Page Observations on the transcendental Equation 2 2 3 2 2 2 2 2 ) 1 ( 2 5 z k Y X x y      M.A.Gopalan 1 , S.Vidhyakshmi 2 , T.R.Usha Rani 3 Department of Mathematics, Shrimati Indira Gandhi College, Tiruchirapalli-2, Abstract: The transcendental equation with five unknowns given 2 2 3 2 2 2 2 2 ) 1 ( 2 5 z k Y X x y      is analysed for its non-zero distinct integral solutions.Various different patterns of integral solutions are illustrated and some interesting relations between the solutions and special numbers are exhibited. Keywords: Surd,transcendental equation,integral points,figurative numbers. Notations: 1 ) 1 ( 6    n n S n -Star number of rank n. )) 1 2 ( ( 3 1 2   n n OH n -Octahedral number of rank n.           2 ) 2 )( 1 ( 1 , m n n t n m -Polygonal number of rank n with size m.   ) 5 ( ) 2 ( 6 ) 1 ( m n m n n P m n      -Pyramidal number of rank n with size m. ) 1 (   n n PR n -Pronic number. 3 6 n CP n   -centered hexagonal pyramidal number of rank n. 2 ) 1 3 ( 2 9   n n CP n - centered Nonagonal pyramidal number of rank n. 6 ) 24 30 ( 2 9   n n CP n - centered Triaconagonal pyramidal number of rank n. 6 ) 2 ( ) 1 ( 2 4 , , 4    n n n F n -Four dimensional Figurative number of rank n whose generating polygon is a square. 6 ) 2 )( 1 ( 2 6 , , 4    n n n F n -Four dimensional Figurative number of rank n whose generating polygon is a pentagon. ! 5 ) 4 )( 3 )( 2 )( 1 ( 3 , , 5      n n n n n F n - Five dimensional Figurative number of rank n whose generating polygon is a triangle. ! 6 ) 5 )( 4 )( 3 )( 2 )( 1 ( 3 , , 6       n n n n n n F n - Six dimensional Figurative number of rank n whose generating polygon is a triangle. I. Introduction Diophantine equations have an unlimited field of research by reason of their variety. Most of Diophantine problems are algebraic equations [1, 2].It seems that much work has not been done to obtain integral solutions of the transcendental equations. In this context, one may refer [4-15]. This communications