Pythagorean TriplesDate: 05/22/99 at 06:04:39 From: Charlie Smith Subject: The general formula for pythagorean triples Hi, I am British and I have been doing some gcse coursework. The project is called "Beyond Pythagoras". I have already found the formula for the shortest, middle, and longest sides of both odd and even triples. I have also found the formulas for the area and perimeter of both the odds and evens, but I am stuck on trying to get the general formula for all sides of any triple. Date: 05/22/99 at 08:10:07 From: Doctor Jerry Subject: Re: The general formula for pythagorean triples Hi Charlie, The following result may be what you're looking for. G. L. Hardy was also British. I don't know about Wright. Pythagorean triples can be generated using the following theorem, taken from Hardy and Wright's _The Theory of Numbers_. The most general solution of the equation x^2+y^2 = z^2 satisfying the conditions x > 0, y > 0, z > 0 x and y have no common factors x is divisible by 2 is x = 2a*b y = a^2-b^2 z = a^2+b^2 where a and b are integers, one even and the other odd, with no common factors, and a > b > 0. There is a 1-1 correspondence between different values of a and b and different values of x,y,z. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 05/24/99 at 08:22:16 From: guy joseph Subject: Pythagorean triples Please could you send me the general formula for all Pythagorean triples, and if possible an explanation, some examples and a proof? Thank you in advance. Date: 05/24/99 at 14:32:00 From: Doctor Anthony Subject: Re: Pythagorean triples Pythagorean Triplets -------------------- We wish to satisfy the Diophantine equation x^2 + y^2 = z^2 If we let x = a^2 - b^2 y = 2ab z = a^2 + b^2 Then x^2 = a^4 - 2a^2.b^2 + b^4 y^2 = 4a^2.b^2 x^2 + y^2 = a^4 + 2a^2.b^2 + b^4 = (a^2 + b^2)^2 = z^2 Thus choosing any pair (a,b) we generate a Pythagorean triplet. However, always choose these such that a > b. Example (2,1) gives x = 4 - 1 = 3 y = 2x2x1 = 4 z = 4 + 1 = 5 Example (3,2) gives x = 9 - 4 = 5 y = 2x3x2 = 12 z = 9 + 4 = 13 Example (3,1) gives x = 9 - 1 = 8 y = 2x3x1 = 6 z = 9 + 1 = 10 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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