# Pythagorean Numbers

## Pythagorean numbers

[pə‚thag·ə′rē·ən ′nəm·bərz] (mathematics)

Positive integers

*x, y,*and*z*which satisfy the equation*x*^{2}+*y*^{2}=*z*^{2}. Also known as Pythagorean triple.McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Pythagorean Numbers

triples of natural numbers such that if the lengths of the sides of a triangle are proportional or equal to the numbers of such a triple, the triangle is a right triangle. By the converse to the Pythagorean theorem, it is sufficient if the numbers satisfy the Diophantine equation *x ^{2} + y^{2} = z^{2}*. An example of such a triple is

*x*= 3,

*y*= 4, and

*z*= 5. All triples of relatively prime Pythagorean numbers can be obtained from the formulas

*x = m ^{2} – n^{2} y = 2mn z = m^{2} + n^{2}*

where *m* and *n* are integers and *m > n* > 0.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.