Useless Numerology for 2016: Part 4

The following entertaining (but useless) facts about the number 2,016 appeared in a recent Facebook post (and subsequent comments) by the American Mathematical Monthly.

$2016 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3$

$2016 = (1+2+...+8+9)^2 - (1+2)^2$

To evaluate the top series, we use the formula

$1^3 + 2^3 + \dots + n^3 = \displaystyle \frac{n^2 (n+1)^2}{4}$

$= \displaystyle \left( \frac{n(n+1)}{2} \right)^2$

$= (1 + 2 + \dots + n)^2$,

where we used the formula for an arithmetic series. Therefore,

$3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = (1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3) - (1^3 + 2^3)$

$= (1 + 2 + \dots + 9)^2 - (1 + 2)^2$.