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): $159$