# 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:

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

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 $2 \times 3^4 = 162 > 159$.

$\blacksquare$

## Also see

- Smallest Number not Expressible as Sum of Fewer than 19 Fourth Powers
- 319 is not Expressible as Sum of Fewer than 19 Fourth Powers

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $159$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $159$