$$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Pythagorean Triples - An Application of Complex Numbers

Solving $a^2+b^2=c^2$ within the set of natural numbers using complex numbers.

Author

Affiliation

Fei Ye

QCC-CUNY

1 Discovering Pythagorean Triples

Recently, I learned a fascinating trick to find Pythagorean triples, i.e., $(a,b,c)$ such that $a^2+b^2=c^2$, from a YouTube video by Professor Richard E. Borcherds. Here’s how it works:

Consider the equation:

$$a+b\mathbf{i}=(x+y\mathbf{i}{)}^2$$

Expanding this, we get:

$$a+b\mathbf{i}=(x^2-y^2)+2xy\mathbf{i}$$

Taking the modulus squared of both sides:

$$\begin{array}{rl}|a+b\mathbf{i}{|}^2& =|(x+y\mathbf{i}{)}^2{|}^2\\ a^2+b^2& =(|x+y\mathbf{i}{|}^2{)}^2\\ a^2+b^2& =(x^2+y^2{)}^2\end{array}$$

By setting $a=x^2-y^2$, $b=2xy$, and $c=x^2+y^2$, we ensure that $a^2+b^2=c^2$.

In other words, we can find Pythagorean triples by choosing integers $x$ and $y$ such that $a=x^2-y^2$, $b=2xy$, and $c=x^2+y^2$.

1.1 Examples of Pythagorean Triples

Example 1  

• For $x=2$ and $y=1$, we get: $$a=2^2-1^2=3,$$ $$b=2\cdot 2\cdot 1=4,$$ $$c=2^2+1^2=5.$$

• For $x=3$ and $y=1$, we get: $$a=3^2-1^2=8,$$ $$b=2\cdot 3\cdot 1=6,$$ $$c=3^2+1^2=10.$$

• For $x=3$ and $y=2$, we get: $$a=3^2-2^2=5,$$ $$b=2\cdot 3\cdot 2=12,$$ $$c=3^2+2^2=13.$$