pythag-triples - Pythagorean Triples Jonathan L.F King...

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Unformatted text preview: Pythagorean Triples Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 26 August, 2007 (at 21:07 ) Entrance. Use “ n ≡ 4 k ” to mean * that 4 •| n- k . A Pythagorean triple h a,b,c i of integers satisfies that a 2 + b 2 = c 2 . Given x,y ∈ Z , define T ( x,y ) := h a,b,c i where a := xy ; b := 1 2 y 2- x 2 ; c := 1 2 y 2 + x 2 . 1 : Evidently T ( x,y ) is a Pythagorean triple. Definition: Primitive triples and good pairs. A prim- itive (Pythagorean) triple h a,b,c i satisfies i : a,b,c ∈ Z + and Gcd( a,b,c ) = 1 . ii : a 2 + b 2 = c 2 . iii : a is odd. Notice that (ii) together with Gcd( a,b,c ) = 1 im- plies that each pair of h a,b,c i is relatively prime. Thus a and b can’t both be even. So there is no loss of generality in the normalization (iii). A pair ( x,y ) is good if x and y are odd, positive integers with 1 6 x < y and x ⊥ y . 2 : Theorem. There is a 1-to-1 correspondence between good pairs and primitive triples: The map GoodPairs T-→ PrimitiveTriples is a bijection. ♦ * Use ≡ N to mean “congruent mod N ”. Let n ⊥ k mean that n and k are co-prime. Use k •| n for “ k divides n ”. Its negation k r | n means “ k does not divide n .” Use n |• k and n r | k for “ n is/is-not a multiple of k .” Finally, for p a prime and E a natnum, have p E •|| n mean that E is the highest power of p which divides n . Or write n ||• p E to emphasize that this is an assertion about n . Our T is well-defined. Firstly, a = xy is odd and a , b and c are positive since 1 6 x < y . Since x 2 and y 2 are each odd, and Odd ± Odd is even, we have that b and c are indeed integers. To establish that h a,b,c i is primitive, what remains is to prove that Gcd( a,b,c ) = 1. So suppose that p is a prime dividing a . Necessarily, p divides x or y ; without loss of generality p •| x . Were p to divide b , forcing p •| [ y 2- x 2 ], then p •| y 2 and consequently p •| y . But that contradicts that ( x,y ) is a good pair....
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

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pythag-triples - Pythagorean Triples Jonathan L.F King...

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