Playing with a pencil and paper notebook I noticed the following:

$x=1$

$x^3=1$

$x=2$

$x^3=8$

$x=3$

$x^3=27$

$x=4$

$x^3=64$

$64-27 = 37$

$27-8 = 19$

$8-1 = 7$

$19-7=12$

$37-19=18$

$18-12=6$

I noticed a pattern for first 1..10 (in the above example I just compute the first 3 exponents) exponent values, where the difference is always 6 for increasing exponentials. So to compute $x^3$ for $x=5$, instead of $5\times 5\times 5$, use $(18+6)+37+64 = 125$.

I doubt I've discovered something new, but is there a name for calculating exponents in this way? Is there a proof that it works for all numbers?

There is a similar less complicated pattern for computing $x^2$ values.