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Unformatted text preview: Pythagorean Triples Is more or less Proposition 29 in Book X of Elements (300 BC). A Pythagorean triple is a triple of numbers (a,b,c) such that a 2 + b 2 = c 2 . A Pythagorean triple is primitive If a , b , and c have no nontrivial common factors. The Theorem: Every primitive Pythagorean triple is of the form ( 2 mn,m 2 n 2 ,m 2 + n 2 ) (or (m 2 n 2 , 2 mn,m 2 + n 2 ) ) for some numbers m > n . Conversely, If m > n are numbers, the formula ( 2 mn,m 2 n 2 ,m 2 + n 2 ) always produces a (not-necessarily primitive) Pythagorean triple. The Proof The Euclidean Algorithm Is more or less Propositions 1 and 2 in Book VII of Elements (300 BC). The Theorem: The greatest common factor of two numbers can be found by repeating the following process: Divide the smaller number into the larger number. If the remainder is zero, the smaller number is the GCD....
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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