Journal of Mathematics and Informatics Vol. 11, 2017, 119-123 ISSN: 2349-0632 (P), 2349-0640 (online) Published 11 December 2017 www.researchmathsci.org DOI: http://dx.doi.org/10.22457/jmi.v11a15 119   Special Dio 3-tuples for Pentatope Number G.Janaki and C.Saranya Department of Mathematics, Cauvery College for Women, Trichy-18, Tamilnadu, India. Received 6 November 2017; accepted 9 December 2017 Abstract. We search for three distinct polynomials with integer coefficients such that the product of any two members of the set added with their sum and increased by a non-zero integer (or polynomial with integer coefficients) is a perfect square. Keywords: Dio 3-tuples, Pentatope number, polynomials. AMS Mathematics Subject Classification (2010): 11D25 1. Introduction Mathematics is the language of patterns and relationships, and is used to describe anything that can be quantified. The main goal of Number theory is to discover interesting and unexpected relationships. It is devoted primarily to the study of natural numbers and integers. In [1-7], theory of numbers were discussed. Many mathematicians considered the problem of the existence of Diophantine triples & special dio 3-tuples with the property ) (n D for any arbitrary integer n and also for any linear polynomials n [8-10]. In this communication, we present a few special dio 3-tuples for Pentatope numbers of different ranks with their corresponding properties. Notation n PT = Pentatope number of rank n = ) 3 ( ) 2 ( ) 1 ( 24 1 + + + n n n n 2. Basic definition A set of three distinct polynomials with integer coefficients ( 3 2 1 , , a a a is said to be a special dio 3-tuple with property ) ( n D if ( 29 n a a a a j i j i + * is a perfect square for all 3 1 ≤ < ≤ j i , where n may be non-zero integer or polynomial with integer coefficients. 3. Method of analysis Case 1: Construction of Dio 3-tuples for Pentatope number of rank n and 2 - n . Let 2 PT 24 , PT 24 - = = n n b a be Pentatope numbers of rank n and 2 - n respectively such that 1 ) ( + + b a ab is a perfect square say 2 α .