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

Contents (PDF format)

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

Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.

[2] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.

[3] Grzegorz Bancerek. Sequences of ordinal numbers. Journal of Formalized Mathematics, 1, 1989.

[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[5] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.

[6] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin's test for the primality of Fermat numbers. Journal of Formalized Mathematics, 10, 1998.

[7] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.

[8] Rafal Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Journal of Formalized Mathematics, 2, 1990.

[9] Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Journal of Formalized Mathematics, 9, 1997.

[10] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.

[11] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[12] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.

[13] Andrzej Trybulec and Czeslaw Bylinski. Some properties of real numbers operations: min, max, square, and square root. Journal of Formalized Mathematics, 1, 1989.

[14] Michal J. Trybulec. Integers. Journal of Formalized Mathematics, 2, 1990.

[15] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[16] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.