# Pythagorean Triples

• May 12th 2009, 11:16 PM
cathwelch
Pythagorean Triples
Question:
Verify that3, 4, 5 is the only primitive Pythagorean triple involving consecutive positive integers.
• May 13th 2009, 04:06 AM
Media_Man
Trouble with Triples
Suppose $a^2+b^2=c^2$ are integers. Without loss of generality, $0.

$x^2+(x+1)^2=(x+2)^2$
$2x^2+2x+1=x^2+4x+4$
$x^2-2x-3=0$
$(x-3)(x+1)=0$
$x=3$ or $x=-1$

But we have already supposed all values are positive, so we discard $(-1,0,1)$ and are left only with $(3,4,5)$.