The Infinitude of Pythagorean Triples

In the Understanding the Fermat’s Last Theorem post, I have mentioned about Pythagorean Triples. In this post, we will show that there are infinitely many of them. We will use intuitive reasoning to prove the theorem.

For the 100th time (kidding), recall that the Pythagorean Theorem states that in a right triangle with side lengths $a, b$ and $c$ , where $c$ is the hypotenuse, the equation $c^2 = a^2 + b^2$ is satisfied. For example, if we have a triangle with side lengths $2$ and $3$ units, then the hypotenuse is $\sqrt{13}$ . The converse of the Pythagorean theorem is also true: If you have side lengths, $a, b$ and $c$ , which satisfies the equation above, we are sure that the angle opposite to the longest side is a right angle.

Are we a Pythagorean Triple or not?

We are familiar with right triangles with integral sides. The triangle with sides $(3, 4, 5)$ units, for instance, is a right triangle. This is also the same with $(5, 12, 13)$ and $(8, 15, 17)$ . We will call this triples, the Pythagorean triples ,or geometrically, right triangles having integral side lengths.

The first thing that we can observe about the Pythagorean triples is that there are infinitely many of them. The triple $(3, 4, 5)$ , for example, can be multiplied by any positive integer to produce another Pythagorean triple. For example $2(3,4,5) = (6,8,10)$ is also a Pythagorean triple. The proof is intuitively discussed below.

Theorem: There are infinitely many Pythagorean Triples.

Proof: We have discussed that two triangles that are similar are of the same shape, but not necessarily of the same size. We also know that similar triangles have congruent corresponding angles. Therefore, if a triangle with integral side lengths $(a, b, c)$ is right, multiplying the side lengths with any positive integer $k$ changes only the size and not the shape. Therefore, our new triangle with side lengths $(ak, bk, ck)$ is also right.

Since there are infinitely many positive integers which we can substitute to $k$ , we can therefore conclude that there are infinitely many Pythagorean Triples.

Now that we have learned that there are infinitely many of them, in the next post, we are going to discuss some interesting strategies in generating Pythagorean triples aside from the one mentioned above.

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Photo Credit: Wikipedia

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