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

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