A Comprehensive Approach to the Pythagorean Triples in Mathematics

A Comprehensive Approach to the Pythagorean Triples in Mathematics

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Category: Mathematics Essay
Subcategory: Geometry
Words: 647
Pages: 4
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Document details
Category: Mathematics Essay
Subcategory: Geometry
Words: 647
Pages: 4

Pythagorean TriplesThree integers a, b, and c that satisfy a2 b2 c2 are called Pythagorean Triples. There are infinitely many such numbers and there also exists a way to generate all the triples. Let n and m be integers, nm. Then define() a n2 - m2, b 2nm, c n2 m2. The three number a, b, and c always form a Pythagorean triple. The proof is simple (n2 - m2)2 (2mn)2 n4 - 2n2m2 m4 4n2m2 n4 2n2m2 m4 (n2 m2)2. The formulas were known to Euclid and used by Diophantus to obtain Pythagorean triples with special properties. However, he never raised the question whether in this way one can obtain all possible triples.The fact is that for m and n coprime of different parities, () yields coprime numbers a, b, and c. Conversely, all coprime triples can indeed be obtained in this manner. All others are multiples of coprime triples ka, kb, kc.As an aside, those who mastered the arithmetic of complex numbers might have noticed that (m in)2 (n2 - m2) i2mn. Which probably indicates that () has a source in trigonometry. But the proof below only uses simple geometry and algebra.First of all, note that if a2 b2 c2, then (ac)2 (bc)2 1. With x ac and y bc we get x2 y2 1. This is the well known equation of the unit circle with center at the origin. Finding Pythagorean triples is therefore equivalent to locating rational points (i.e., points (x,y) for which both x and y are rational) on the unit circle. For if (pq)2 (rs)2 1, multiplication by a common denominator leads to an identity between integers.Rational numbers approximate irrational to any degree of accuracy. Therefore, the set of rational pairs is dense in the whole plane. So, perhaps, one might...

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