A pythagorean triple consists of three positive integers such that $c^2=a^2+b^2$. The three integers 3, 4 and 5 is a well known pythagorean triple because $ 5^2=3^2+4^2 $.
A pythagorean triple is easy to work with when training math because you don´t have to use rational numbers, only integers, and therefore loved by math teachers 🙂
(3, 4, 5)
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)
(20, 21, 29)
(12, 35, 37)
(9, 40, 41)
(28, 45, 53)
This is only a small list since it exists an infinite amount of pythagorean triples.
You can actually fairly simple create your own triples by scaling up one set. You do this by multiplying each value of the triple with a positive integer.
For example the pythagorean triple (3, 4, 5) can be multiplied with 3:
$(3·3, 4·3, 5·3) = (9, 12, 15)$
Let´s check if the pythagorean theorem still holds:
$ 9^2+12^2= 225$
$ 15^2=225 $
The both sides equals each other!
Let´s check if it holds when we use algebra.
Let´s assume that we have found a triple $(a,b,c)$ where $a^2 + b^2 = c^2 $
Multiply the triple $(a,b,c)$ with a constant k so that we get $ (k·a, k·b, k·c) $
We now want to prove that the following holds
$(k·a)^2+(k·b)^2 = (k·c)^2 $
Expand both sides
$k^2a^2 +k^2b^2 = k^2c^2 $
$k^2(a^2 + b^2) =k^2c^2 $
If we divide both sides with $k^2$ we get
$a^2 +b^2 = c^2$ which we assumed holds from the beginning.
So the left side equals the right side and the proof is done.