But the margin is too narrow to contain!

Geometry Level 3

Let $a,b,c$ be positive integers. Which of the following combination can possibly be the side lengths of a right triangle?

\big(a^2,b^2,c^2\big)

\big(a^3,b^3,c^3\big)

\big(a^4,b^4,c^4\big)

None of them

1 solution

Marco Brezzi
Jul 31, 2017

Since the choices are symmetrical in $a,b,c$ , wlog $c\geq a\geq b$

If the choices are the sides of a right triangle, this translates to

$a^4+b^4=c^4$

$a^6+b^6=c^6$

$a^8+b^8=c^8$

But since $a,b,c\in\mathbb{Z}^+$ and the exponents are equal and bigger than $2$ , by Fermat's Last Theorem, there exist no such $a,b,c$