Euler's conjecture (1769) states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a k-th power, then n is greater than or equal to k.
For k=2 (3,4,5) and for k=3 (3,4,5,6) support this conjecture,but it was later proven false for k=4 and 5.
To find 5 integers such that a^4+b^4+c^4=d^4 is extremely difficult (big,big numbers), but for fellows floobers to solve
a^5+b^5+c^5+d^5=e^5
is a piece of cake, since (HINT) all the integers are below 200(sic!).
OK, it is prestigious, albeit easy task:
Defeat Euler for k=5 .