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

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