Question:

Verify that3, 4, 5 is the only primitive Pythagorean triple involving consecutive positive integers.

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- May 12th 2009, 10:16 PMcathwelchPythagorean Triples
Question:

Verify that3, 4, 5 is the only primitive Pythagorean triple involving consecutive positive integers. - May 13th 2009, 03:06 AMMedia_ManTrouble with Triples
Suppose $\displaystyle a^2+b^2=c^2$ are integers. Without loss of generality, $\displaystyle 0<a<b<c$.

$\displaystyle x^2+(x+1)^2=(x+2)^2$

$\displaystyle 2x^2+2x+1=x^2+4x+4$

$\displaystyle x^2-2x-3=0$

$\displaystyle (x-3)(x+1)=0$

$\displaystyle x=3$ or $\displaystyle x=-1$

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