# Pythagorean Triple 2: Generating Pythagorean Triples

In the previous math article, we have shown that there are infinitely many Pythagorean triples.  In this article, we are going to discuss a very short but effective strategy in generating Pythagorean Triples.

A Pythagorean triple is the integer triple $(a,b, c)$ satisfying the Pythagorean equation $a^2 + b^2 = c^2$.

Observe the Pythagorean triples $(3, 4, 5)$ and  $(5, 12, 13)$.  We can see that the hypotenuse is greater than the longer side by $1$. From the pattern, we can form the Pythagorean triples $(a, b, b + 1)$ satisfying the equation $a^2 + b^2 = (b + 1)^2$.

Right triangle with side length 3, 4 and 5 units.

Solving the equation we have $a^2 = 2b + 1$, which implies that  $a = \sqrt{2b +1}$.  Now,   $2b + 1$ is always odd (can you see why?). It follows that in order for $a$ to be an integer, $2b + 1$ must be a perfect square. This means, that we are sure that $a$ is an integer, if $2b + 1$ is an odd perfect square.

From here, we can generate infinitely many examples of Pythagorean triples.  For example, $49$ is an odd perfect square. So plugging it in the equation we have,  $2b + 1 = 49$, then, we have the triple $(7, 24, 25)$, another Pythagorean Triple.  If we let $2b + 1 = 121$, then we have the triple $(11, 60, 61)$.

Now, we found another way to generate Pythagorean Triples.

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## 4 thoughts on “Pythagorean Triple 2: Generating Pythagorean Triples”

1. This is super neat! I covered Pythagoren Triples in geometry this year and there wasn’t much I did with them; I just explained what they are and how they can be recognized. My students didn’t seem to dislike Pythagorean Triples but they didn’t seem to understand or feel a purpose for them either. I think that it would be neat to explore this with students and see if they could notice this pattern and generate a way to find Pythagorean Triples!