Journal of Mathematics

Volume 2019, Article ID 4286517, 8 pages

https://doi.org/10.1155/2019/4286517

## Pythagorean Triples with Common Sides

^{1}Strathmore Institute of Mathematical Sciences, Nairobi, Kenya^{2}Department of Biometry and Mathematics, Botswana University of Agriculture and Natural Resources, Gaborone, Botswana

Correspondence should be addressed to Raymond Calvin Ochieng; ude.eromhtarts@gneihcor

Received 16 January 2019; Accepted 6 March 2019; Published 1 April 2019

Academic Editor: Ji Gao

Copyright © 2019 Raymond Calvin Ochieng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

There exist a finite number of Pythagorean triples that have a common leg. In this paper we derive the formulas that generate pairs of primitive Pythagorean triples with common legs and also show the process of how to determine all the primitive and nonprimitive Pythagorean triples for a given leg of a Pythagorean triple.

#### 1. Introduction

A Pythagorean triple (PT) is a triple of positive integers , which satisfy the Pythagorean equation where represents the length of the hypotenuse; and represent the lengths of the other two sides (legs) of a right triangle. We say a Pythagorean triple is primitive if the numbers , , and are pairwise coprime (see [1]).

Several methods have been formulated that generate Pythagorean triples (see [2–9]). For instance, the most common one is the classical Euclid formula [10, 11]: whenever . A triple generated by this method is primitive if and only if and is odd. Note that is odd if , have opposite parity [12], or

Now by Euclid’s general formula, any Pythagorean triple, primitive and nonprimitive triples, can be written as , where is some positive integer and , are as defined in [10, 11].

Pythagorean triples form different patterns that can be classified and applied in various fields such as cryptography; see [13–20]. For instance, there exist Pythagorean triples that have identical legs; e.g., and are two primitive triples with , while and are nonprimitive triples with the same leg. Similarly , , , and are four primitive Pythagorean triples which have as the identical leg. The nonprimitive triples which share the leg are discussed in Example 1.

In [1], Sierpinski states and proves that there exist only a finite number of Pythagorean triples with a given leg He further states and proves that, for each positive integer , there exist at least different Pythagorean triples with the same leg , where For instance, if we take where , then we obtain Pythagorean triples with the same leg and with different hypotenuses.

It is also stated in [1] that it is not easy to prove that, for each positive integer , there exist at least different primitive Pythagorean triples with an identical leg. However in this paper we prove formula for determining pairs of primitive Pythagorean triples which have identical legs. We also show how to determine all the primitive and nonprimitive Pythagorean triples which have a given identical leg.

#### 2. Pairs of Pythagorean Triples with Identical Leg

Consider the following pairs of primitive Pythagorean triples which can be generated from the equation

From Table 1, we observe that, for each pair of triples with identical leg, the difference between the hypotenuse and the odd leg is either or We then have Solving (5), we obtain Observe from Table 1 that is the form where Substitute this to obtain In a similar way, we can obtain and from (6).