Can the Pythagorean Theorem be extended like this?

Question

What is the significance, if any, of the fact that

$3^3$ + $4^3$ + $5^3$ = $6^3$

?

How curious this is. This would be like the Pythagorean Theorem exploding into a higher dimension, on steroids.

Answer 1

This is, in fact, the analogous expression to the distance between two points $x$ and $y$ in the space $\Bbb{R}^4$. However, the "euclidean" (or "pythagorean") distance in $n$ dimensions is given by: $$\sqrt{\sum_{i=0}^{n} (x_i - y_i)^2} $$ There is a generalization, called Minkowski distance: $$\Big({\sum_{i=0}^{n} |x_i - y_i|^{1/p}}\Big)^p$$ And you recognise here your expression with $p=1/3$.

In conclusion, this is an integer expression for the distance between two points in $\Bbb{R}^4$, with the $1/3$-Minkowski metric.