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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 6, No. 3, 2013, 363-364

ISSN 1307-5543 – www.ejpam.com

Dickson’s Method for Generating Pythagorean Triples

Revisited

Josef Rukavicka

Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague

Abstract. The Dickson’s method for generating Pythagorean triples states that the integers a = r + s,

b = r + t, c = r + s+ t form a Pythagorean triple (a, b, c) on condition that r2 = 2st, where r, s, t are

positive integers. This paper presents a new simple proof of this method.

2010 Mathematics Subject Classifications: 11D45

Key Words and Phrases: Pythagorean triples, Dickson’s method, Diophantine approximation

1. Introduction

A triple (a, b, c) of positive integers is called a Pythagorean triple in case that a2+ b2 = c2.

There are several methods for generating Pythagorean triples. In this paper we investigate the

Dickson’s method, [1], that states that the integers a = r + s, b = r + t, c = r + s+ t form a

Pythagorean triple (a, b, c) on condition that r2 = 2st, where r, s, t are positive integers. The

paper presents a new simple proof of this method.

2. Dickson’s Method for Generating Pythagorean Triples

Theorem 1. Given positive integers k,q, p, where k > q and let c = k + p, b = p + q, a = k.

Then a2+ b2 = c2 if and only if q2 = 2p(k− q).

Proof. We will show a bijection between triples (k,q, p) and (a, b, c). The bijection is

depicted in Figures 1a and 1b (there are two examples for k = 8, q = 4, p = 2, a = k = 8,

b = p + q = 6, c = k + p = 10 and k = 8, p = 9, q = 6, a = k = 8, b = p + q = 15,

c = k+ p = 17).

Email addresses: josef.rukavicka@seznam.cz, rukavij@fel.cvut.cz

http://www.ejpam.com 363 c© 2013 EJPAM All rights reserved.



REFERENCES 364

(a) q = 4 and p = 2

(b) q = 6 and p = 9

Figure 1: Examples of bijections between triples with k = 8 varying q and p.

The bijection consists in transforming the grid of k × k squares into an object that uniquely

determines a Pythagorean triple (a, b, c). The crucial observation is that q2 is divisible by

2(k−q) and hence the squares from the grid of q2 squares can be equally separated in order to

build up the L-shaped object on the right side of a figure; that object represents a Pythagorean

triple (k,q+ p, k+ p) = (a, b, c). To see this just note the L-shaped object is composed of k2

squares and adding (p+ q)2 squares produces (k+ p)2 squares. On the other hand, given an

L-shaped object composed of k2 squares, then k,q, p are uniquely determined.

Also note that (b, a, c) and (a, b, c) will yield different values of (k, p,q).

References

[1] L.E. Dickson. History of the theory of numbers, vol. 2: Diophantine analysis. Carnegie

Institution of Washington, Publication No. 256, 2:169, 1920.