International Journal of Mathematics and Mathematical Sciences

Volume 2016, Article ID 5189057, 6 pages

http://dx.doi.org/10.1155/2016/5189057

## Almost and Nearly Isosceles Pythagorean Triples

Department of Mathematics, Han Nam University, Daejeon, Republic of Korea

Received 16 June 2016; Accepted 9 August 2016

Academic Editor: Aloys Krieg

Copyright © 2016 Eunmi Choi. 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

This work is about extended pythagorean triples, called NPT, APT, and AI-PT. We generate infinitely many NPTs and APTs and then develop algorithms for infinitely many AI-PTs. Since AI-PT is of , we ask generally for PT satisfying for any . These triples are solutions of certain diophantine equations.

#### 1. Introduction

A pythagorean triple (PT) is an integer solution satisfying the polynomial , and it is said to be primitive (PPT) if . There have been many ways for finding solutions of , and one of the well-known methods is due to Euclid, BC 300. The investigation of integer solutions of has been expanded to various aspects. One direction is to deal with polynomials , where in [1] its integer solutions were called almost pythagorean triple (APT) or nearly pythagorean triple (NPT) depending on the sign . Another side is to study solutions of having some special conditions. A solution is called isosceles if . Since there is no isosceles integer solution of , isosceles-like integer triples with were investigated. We shall call the an almost isosceles pythagorean triple (AI-PT), and typical examples are and . In literatures [2–4], AI-PT was studied by solving Pell polynomial. And a few others [5, 6] used triangular square numbers for finding AI-PT. We note that in some articles AI-PT was called almost isosceles right angled (AIRA) triangle. But in order to emphasize relationships with PT, APT, and NPT in this work, we shall refer to AIRA as AI-PT. APT and NPT were studied in [1] while AI-PT was studied in [2, 4], and so forth, but it seems that no one has asked about their connections.

In this work we generate infinitely many APTs and NPTs and then apply the results in order to develop algorithms for constructing infinitely many AI-PTs. Moreover we study PTs satisfying for any . So the study of these triples can be regarded as a research of solving diophantine equations and .

#### 2. Almost and Nearly Pythagorean Triples

APT and NPT, respectively, are integer solutions of and , respectively. If is an APT or NPT, so it is hence we generally assume . Some triples were listed in [1] by experimental observations: NPT: , , , APT: , , , , ,

Lemma 1 (see [1]). *If is an APT then is a NPT. Conversely if is a NPT then is an APT.*

Theorem 2. *If is an even integer then we have the following. *(1)* is an APT if , while it is a NPT if .*(2)* is an APT and is a NPT.*

*Proof. *If then . If then , so is an APT. If then , so is a NPT.

Due to Lemma 1, the NPT yields an APT , while the APT provides a NPT (see Table 1).