# Math Help - Pythagorean Triples

1. ## Pythagorean Triples

Find all of the Pythagorean triples that consist of consecutive positive integers. Prove that you have found them all.

2. Hello, MATNTRNG!

Find all of the Pythagorean triples that consist of consecutive positive integers.
Prove that you have found them all.

Let the triple be: . $(a,\:a+1,\:a+2)$

Then we have: . $a^2 + (a+1)^2 \;=\;(a+2)^2$

. . . . . . . . . $a^2 + a^2 + 2a + 1 \;=\;a^2+4a+4$

. . . . . . . . . . . . $a^2 - 2a - 3 \:=\:0$

. . . . . . . . . . $(a+1)(a-3) \:=\:0$

. . . . . . . . . . . . . . . . . . . $a \:=\:\text{-}1,\:3$

Since $\,a$ is a positive integer, $a = 3.$

Therefore, the only Pythagorean triple is: . $(3,\,4,\,5)$