Numbers not Expressible as Sum of Fewer than 19 Fourth Powers

Theorem
The following positive integer are the only ones which cannot be expressed as the sum of fewer than $19$ fourth powers:


 * $79, 159, 239, 319, 399, 479, 559$

Proof
On a case-by-case basis:

From Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers:

From 159 is not Expressible as Sum of Fewer than 19 Fourth Powers:

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

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

From 399 is not Expressible as Sum of Fewer than 19 Fourth Powers

From 479 is not Expressible as Sum of Fewer than 19 Fourth Powers

From 559 is not Expressible as Sum of Fewer than 19 Fourth Powers

Also see

 * Hilbert-Waring Theorem