@IJMTER-2016, All rights Reserved 287 GAUSSIAN INTEGER SOLUTIONS TO SPACE PYTHAGOREAN EQUATION Dr. Manju Somanath 1 , J. Kannan 2 and Mr. K. Raja 3 1 Assistant Professor, Department of Mathematics, National College 2 Research Scholar, Department of Mathematics, National College 3 Assistant Professor, Department of Mathematics, National College Abstract – Gaussian integer solutions of the Space Pythagorean equation are obtained. Keywords – Gaussian integer, Space Pythagorean 4 – tuples, Diophantine equation, integral solutions, Pythagorean equation. I. INTRODUCTION Number theory is the branch of mathematics concerned with studying the properties and relations of integers. There are number of branches of number theory of which Diophantine equation is very important. Diophantine equations are numerically rich because of their variety [1, 2, 3]. One of the most important Diophantine equation since antiquity till today is the Space Pythagorean equation . Different patterns of integer solution to this equation is discussed by various authors [4, 5, 6]. In this paper, we search for Gaussian integer solutions to Space Pythagorean equation. Different patterns of solutions are also obtained. II. METHOD OF ANALYSIS The equation to be solved is (1) The different patterns of solutions to (1) are presented below: The substitution , , , (2) in (1), leads to ( ) +( ) ( ) =( ) (3) Equating the real and imaginary parts, we get (4) (5) Taking () becomes √ (6) Assuming we have (7) A. Case (I) (7) can be solved as (8) Substituting these values in (7) we get two choices of Choice 1: , Hence substituting this value in (6) and using (4) and (2), we obtain integer solutions to (1) as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )