Journal of Formalized Mathematics
Volume 13, 2001
University of Bialystok
Copyright (c) 2001 Association of Mizar Users

## Pythagorean Triples

Freek Wiedijk
University of Nijmegen

### Summary.

A Pythagorean triple is a set of positive integers $\{ a,b,c \}$ with $a^2 + b^2 = c^2$. We prove that every Pythagorean triple is of the form $$a = n^2 - m^2 \qquad b = 2mn \qquad c = n^2 + m^2$$ or is a multiple of such a triple. Using this characterization we show that for every $n > 2$ there exists a Pythagorean triple $X$ with $n\in X$. Also we show that even the set of {\em simplified\/} Pythagorean triples is infinite.

#### MML Identifier: PYTHTRIP

The terminology and notation used in this paper have been introduced in the following articles [11] [15] [5] [3] [12] [7] [14] [8] [1] [13] [9] [6] [10] [16] [4] [2]

#### Contents (PDF format)

1. Relative Primeness
2. Squares
3. Distributive Law for HCF
4. Unbounded Sets are Infinite
5. Pythagorean Triples

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