The entry for 29 in The Penguin Dictionary of Curious and Interesting Numbers by David Wells (1987) contains:
No sum of three 4th powers is divisible by either 5 or 29 unless they all are. [Euler]
1. If three aren't enough, how many 4th powers does it take to be divisible by either 5 or 29 where they aren't all?
2. If possible, find the next number beyond 5 and 29 that does not divide a sum of three 4th powers.
3. Prove every even number takes at most two 4th powers.
For example using 18 we have 34+154 = 50706 = 18*2817
4. What is the largest number of 5th powers whose sum is divisible by a number N where they aren't all divisible by N?
5. Prove that for higher powers, there is no limit to how many numbers it can take.