Is it possible that in a Pythagorean triple only 2 values in (a, b, c) share a common factor greater than 1.

by Rakend Chauhan   Last Updated July 12, 2019 07:20 AM

In a Primitive Pythagorean Triple (PPT) the set $(a, b, c)$ has GCD of 1.

I have read in the wiki page that every other Pythagorean triple can be obtained by multiplying k to this set $(ka, kb, kc)$ where $k$ is some positive integer.

I was wondering if the above statement is true then it's not possible for only 2 values in any Pythagorean triple to have a common factor greater than 1. Since the common factor will be forced to divide even the third value.

If I am not wrong in asking the question taking this equation

$a^2 + b^2 = c^2$

how to prove the above fact?

Answers 1

If $d=(a,b),\dfrac aA=\dfrac bB=d$(say)

$$c^2=d^2(A^2+B^2),\dfrac{c^2}{d^2}=A^2+B^2$$ which is an integer

$\implies d^2\mid c^2\implies d\mid c$

Similarly if $e=(a,c)$

lab bhattacharjee
lab bhattacharjee
July 12, 2019 07:09 AM

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