Pythagorean Triples

Date: 04/14/97 at 09:05:48
From: Paul W. Paulsen
Subject: Pythagorean Triples

At a recent Parent-Teacher Conference Night a parent asked me why, 
in a Pythagorean triple (e.g. 3,4,5), the three numbers cannot all 
be prime.  Is this possible?  Example?  Why not?

Thanks.
Paul Paulsen

Date: 04/14/97 at 10:25:08
From: Doctor Mitteldorf
Subject: Re: Pythagorean Triples

Dear Paul,

If you think "prime" it makes it sound like a high-falutin' result 
from number theory.  Think "even" and "odd" instead.

If a^2 + b^2 = c^2, then a, b, and c cannot all be odd.  If a and b 
are odd, then c is even, as it is the square root of the sum of two 
odd numbers.

This leaves only cases where a = 2 or b = 2, since 2 is the only even 
prime.  Notice that squares get increasingly far apart, so 1 and 4 
differ by 3, but every other pair of squares differs by at least 5.  

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/