Is there a pattern in Pythagorean Triples?

Let's look at the first 5 Pythagorean Triples:

3-4-5, 5-12-13, 7-24-25, 9-40-41, 11-60-61

Are you noticing a pattern here?
The first number in each triple is just 3, 5, 7, 9, 11, adding 2 each time.
The second number in each triple is 4, then 12 (adding 8), then 24 (adding 12), then 40 (adding 16), then 60 (adding 20). You add 4 to the number you add to the number every time.
The third number in each triple is always just the second number + 1.
Here is a Python 3.4 algorithm to simulate this (sorry for bad quality).
By this algorithm, the 1,000,000th Pythagorean triple is (drumroll please) 2,000,001-2,000,002,000,000-2,000,002,000,001!
Please let me know if you can find a proof for this algorithm.

#NumberTheory

Easy Math Editor

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Comments

Look up Euclid's formula. In your case, you are increasing $m$ by a value of 1 at every step, and you set $n=1$ .

Pi Han Goh - 3 years, 1 month ago

thank you!

Stefan Popescu - 3 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$