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### Pythagorean Triples

```
Date: 11/19/97 at 22:41:34
From: matt
Subject: Pythagorean triples

I need to know the first five Pythagorean triples after 3,4,5. I think
I have the first three: (20,21,29) (119,120,169) and (696, 697, 985).

I also need a formula to derive Pythagorean triples. Thanks!
```

```
Date: 11/20/97 at 07:30:06
From: Doctor Anthony
Subject: Re: Pythagorean triples

Basically you are looking for integer solutions to the 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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory

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