Smallest Fourth Power which is Sum of 5 Fourth Powers
Jump to navigation
Jump to search
Theorem
$625$ is the smallest fourth power which is the sum of $5$ fourth powers:
- $625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$
Proof
We check that for $n = 2, 3, 4$, $n^4$ is not a sum of $5$ smaller fourth powers.
We have:
- $5 \times 1^4 = 5 < 16 = 2^4$
- $5 \times 2^4 = 80 < 81 = 3^4$
so $2^4, 3^4$ are not sums of $5$ fourth powers.
For $n = 4$:
- $\dfrac {4^4} {3^4} < 4$
so such a sum can include at most $3$ $3^4$'s.
However:
- $3 \times 3^4 + 2^4 + 1^4 = 260 > 256 = 4^4$
- $3 \times 3^4 + 2 \ \ \times 1^4 = 245 < 256 = 4^4$
- $2 \times 3^4 + 3 \ \ \times 2^4 = 220 < 256 = 4^4$
therefore $4^4$ is not a sum of $5$ smaller fourth powers.
This shows that $5^4$ is the smallest fourth power which is the sum of $5$ fourth powers.
$\blacksquare$
Also see
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $625$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $625$