Pythagorean Triples as told by Prof. Ian Stewart

Ref: Professor Stewart’s Cabinet of Mathematical Curiosities: Ian Stewart

Pythagorean Triples:

I can’t really get away without telling you Diophantus’s method for finding all Pythagorean triples, can I?

OK, here it is. Take any two whole numbers, and form:

twice their product
the difference between their squares
the sum of their squares

Then, the resulting three numbers are the sides of a Pythagorean triangle.

For instance, take the numbers 2 and 1. Then,

twice their product = $2 \times 2 \times 1=4$
the difference between their squares = $2^{2}-1^{2}=3$
the sum of their squares = $=2^{2}+1^{2}=5$

and, we obtain the famous 3-4-5 triangle. If instead we take numbers 3 and 2, then

twice their product = $2 \times 3 \times 2 = 12$
the difference between their squares = $3^{2}-2^{2}=5$
the sum of their squares = $3^{2}+2^{2}=13$

and, we get to the next-most-famous 5-12-13 triangle. Taking numbers 42 and 23, on the other hand, leads to

twice their product = $2 \times 42 \times 23=1932$
the difference between their squares = $42^{2}-23^{2}=1235$
the sum of their squares = $42^{2}+23^{2}=2293$

and no one has ever heard of the 1235-1932-2293 triangle. But these numbers do work:

$1235^{2}+1932^{2}=1525225+3732624=5257849=2293^{2}$ .

There’s a final twist to Diophantus’s rule. Having worked out the three numbers, we can choose any other number we like and multiply them all by that. So, the 3-4-5 triangle can be converted to a 9-12-15 triangle by multiplying all three numbers by 3, or to an 18-24-30 triangle by multiplying all three numbers by 6. We can’t get these two triples from the above prescription using whole numbers, Diophantus knew that.

More later,

Nalin Pithwa

This entry was written by Nalin Pithwa, posted on February 7, 2016 at 11:27 am, filed under Fun with Mathematics and tagged diophantus. Bookmark the permalink. Follow any comments here with the RSS feed for this post. Post a comment or leave a trackback: Trackback URL.

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