## Problem 1

Find all sets of 14 or less fourth powers which sum to 1599 .


## Solution

Answer: none.

The only 4th powers less than 1599 are $1,16,81,256,625,1296\left(7^{4}=2401\right)$. Note that $1,81,625=1 \mathrm{mod}$ 16 and $16,256,1296=0 \bmod 16$. But $1599=15 \bmod 16$, so it cannot be expressed as a sum of 14 or less fourth powers.

