SELF GENERATING n-TUPLES

Abstract

Background: The Pythagorean triple based on Pythagorean Theorem, were known in to ancient Babylon and Egypt. The interrelation of the three was known as far back as thousands of years, but it was Pythagoras who explicitly explained their equation.
Purpose: Different methods have been put forth by the mathematicians for generation of Pythagorean’s triple and n-tuples but this paper provides a unique method how these get self-generated by use of simple algebraic expansions.
Methods: An algebraic quantity (a+b) squared equals to (a-b) squared plus 4ab and if a or b is assigned such a value that makes 4ab a whole square, then (a+b), (a-b) and under root of 4ab turns Pythagorean’s triple. Similarly, utilizing such algebraic identities, Pythagorean’s quadruple up to n-tuples can be generated. If (a+b) is squared, it provides a squared plus b squared plus 2ab. If quantity 2ab is transformed to a whole square on account of assigning values to a or b, then Pythagorean’s quadruples are obtained. 
Results: Assigning specific values to the terms of simple algebraic identities results in the generation of Pythagorean triples and n-tuples.
Conclusions: This paper presents empirical research in which algebraic identities are utilized, resulting in the self-generation of Pythagorean n-tuples. Specific formulas need not be applied, as basic algebraic identities are well known to scholars and students alike.

Keywords: Pythagorean’s triples, Quadruples, Quintuples