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

Theorem

$319$ cannot be expressed as the sum of fewer than $19$ fourth powers:

$319 = 15 \times 1^4 + 3 \times 2^4 + 4^4$

or:

$319 = 12 \times 1^4 + 4 \times 2^4 + 3 \times 3^4$

First note that $5^4 = 625 > 319$.

Then note that $2 \times 4^4 = 512 > 319$.

Hence any expression of $319$ as fourth powers uses no $n^4$ for $n \ge 5$, and uses not more than $1$ instance of $4^4$.

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

Now we have:

$\ds 319$

$=$

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

$\ds $

$=$

$\ds 3 \times 3^4 + 4 \times 2^4 + 12 \times 1^4$

$\ds $

$=$

$\ds 2 \times 3^4 + 9 \times 2^4 + 13 \times 1^4$

$\ds $

$=$

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

$\ds $

$=$

$\ds 0 \times 3^4 + 19 \times 2^4 + 15 \times 1^4$

and it can be seen that the first two use the least number of fourth powers, at $19$.

$\blacksquare$