Smallest Fourth Power which is Sum of 5 Fourth Powers

Theorem

$625$ is the smallest fourth power which is the sum of $5$ fourth powers:

$625 = 5^4 = 2^4 + 2^4 + 3^4 + 4^4 + 4^4$

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.

$\blacksquare$