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

Theorem

$159$ cannot be expressed as the sum of fewer than $19$ fourth powers:

$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$

We have:

$4^4 = 256 > 159$

$3^4 = 81$

$2^4 = 16$

$1^4 = 1$

Let us attempt to construct an expression of $159$ as the sum of fewer than $19$ fourth powers:

If no $3^4$ is used in our sum, the sum consists only of $2^4$ and $1^4$.

Using $2^4$ is more efficient than using $1^4$, since $2^4$ can replace $16 \times 1^4$.

So we have:

$159 = 9 \times 2^4 + 15 \times 1^4$

which uses $24$ fourth powers.

If $3^4$ is used in our sum, the most efficient way is demonstrated above:

$159 = 14 \times 1^4 + 4 \times 2^4 + 3^4$

which uses $19$ fourth powers.

We cannot use more than $2$ instances of $3^4$, as $2 \times 3^4 = 162 > 159$.

$\blacksquare$