Square of Hypotenuse of Pythagorean Triangle is Difference of two Cubes/Refutation
Theorem
Let $h$ be the hypotenuse of a Pythagorean triangle.
Then it is not necessarily the case that:
- $h^2 = a^3 - b^3$
for some $a, b \in \Z_{>0}$.
Refutation
Consider the equation:
- $(1): \quad h^2 = a^3 - b^3: a, b \in \Z_{>0}$
$h$ itself cannot be a cube, as this would be a counterexample to Fermat's Last Theorem.
Suppose we relax $\Z_{>0}$ to $\Z_{\ge 0}$.
It can be shown that $(1)$ does not hold even among the primitive Pythagorean triples.
We demonstrate that there is no solution for $h = 5$:
Aiming for a contradiction, suppose there is.
We must have $\paren {a - b} \divides h^2$.
For $x \ge h$:
- $\paren {x + 1}^3 - x^3 = 3 x^2 + 3 x + 1 > h^2$
Hence $a \le 5$.
Combined with our divisibility condition, we must have:
- $a - b = 1$ or $5$
But:
- $2^3 < 25$
- $3^3 - 2^3 = 19$
- $4^3 - 3^3 > 25$
- $5^3 - 0^3 > 25$
So there is no solution for $a^3 - b^3 = 25$.
$\blacksquare$
Historical Note
The statement:
- The square of the hypotenuse of a right-angled triangle is also the difference of $2$ cubes; thus, $13^2 = 8^3 - 7^3$.
appears in the $1997$ edition of Curious and Interesting Numbers, 2nd ed. by David Wells.
There is no record of where he may have found this statement, but it is demonstrably false.