Smallest Fourth Power as Sum of 5 Distinct Fourth Powers

Theorem
The smallest $4$th power which can be expressed as the sum of $5$ distinct $4$th powers is:
 * $15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4$

Proof
By Fermat's Little Theorem, for $5 \nmid a$:
 * $a^4 \equiv 1 \pmod 5$

For $5 \divides a$:
 * $a^4 \equiv 0 \pmod 5$

Therefore in order for the equality to hold, one of the following must be true:

The $4$th power on the is not divisible by $5$, while exactly $1$ of the $5$ $4$th powers on the  is not divisible by $5$; or

The $4$th power on the is divisible by $5$, while all of the $5$ $4$th powers on the  is divisible by $5$; or

The $4$th power on the is divisible by $5$, while none of the $5$ $4$th powers on the  is divisible by $5$.

In the first two cases, there must be at least $4$ distinct $4$th powers divisible by $5$.

The smallest multiples of $5$ are $0, 5, 10, 15$, so the $4$th powers on the must exceed $15^4$, and thus these cases cannot produce a smaller example.

This leaves us with the third case, where is divisible by $5$.

Since $5^4 \ne 4^4 + 3^4 + 2^4 + 1^4 + 0^4$, we only need to check that $10^4$ is not a sum of $5$ distinct $4$th powers.

As $8^4 + 7^4 + 6^4 + 5^4 + 4^4 = 8674 < 10^4$, we must have $9^4$ on the.

As $9^4 + 8^4 = 10657 > 10^4$, we cannot have $8^4$ on the.

As $9^4 + 6^4 + 5^4 + 4^4 + 3^4 = 8819 < 10^4$, we must have $7^4$ on the.

As $9^4 + 7^4 + 6^4 = 10258 > 10^4$, we cannot have $6^4$ on the.

But $9^4 + 7^4 + 5^4 + 4^4 + 3^4 = 9924 < 10^4$.

Therefore $10^4$ is not a sum of $5$ distinct $4$th powers.

This proves that $15^4$ is the smallest $4$th power which can be expressed as the sum of $5$ distinct $4$th powers.

Also see

 * Smallest Fourth Power which is Sum of 5 Fourth Powers