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

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Theorem

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

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


Proof

We have:

$4^4 = 256 > 159$, $3^4 = 81$, $2^4 = 16$, $1^4 = 1$.

We 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 only consists 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 $2$ $3^4$ since $2 \times 3^4 = 162 > 159$.

$\blacksquare$


Also see


Sources