239 is not Expressible as Sum of Fewer than 19 Fourth Powers

Theorem

$239$ cannot be expressed as the sum of fewer than $19$ fourth powers:

$239 = 13 \times 1^4 + 4 \times 2^4 + 2 \times 3^4$

First note that $4^4 = 256 > 239$.

Then note that $3 \times 3^4 = 243 > 239$.

Hence any expression of $239$ as fourth powers uses no $n^4$ for $n \ge 4$, and uses not more than $2$ instances $3^4$.

For the remainder, using $2^4$ uses fewer fourth powers than $16$ instances $1^4$ does.

Now we have:

$\ds 239$

$=$

$\ds 2 \times 3^4 + 4 \times 2^4 + 13 \times 1^4$

$\ds $

$=$

$\ds 1 \times 3^4 + 9 \times 2^4 + 14 \times 1^4$

$\ds $

$=$

$\ds 0 \times 3^4 + 13 \times 2^4 + 15 \times 1^4$

and it is apparent the first one uses the least number of fourth powers, at $19$.

$\blacksquare$