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

Proof
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:

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

Also see

 * Hilbert-Waring Theorem


 * Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers
 * 159 is not Expressible as Sum of Fewer than 19 Fourth Powers
 * 399 is not Expressible as Sum of Fewer than 19 Fourth Powers