Euler's sum of powers conjecture generalizes Fermat's last theorem:
\( \sum_{k=1}^m (a_k)^n = b^n \)
is only possible for integer a's and b if m >= n where n > 2.
For n = 3, we get the cube case of Fermat's last theorem, while for n = 4 and 5, we have counterexamples. Here are the smallest ones:
95800^4 + 217519^4 + 414560^4 = 422481^4
27^5 + 84^5 + 110^5 + 133^5 = 144^5
The first one was found in 1988, and the second one in 1966 on a CDC 6600 computer.
I verified the latter one by brute force on my iMac. Its CPU is a 2.3-GHz Intel Core i5 chip. Its optimizations were compiler optimizations, generating semi-increasing sequences directly, and precalculating all the power values. I used C++ STL binary_search to check on whether a sum value was also a power value. I checked up to 150, checking through 22 million sets of values, and it took 0.4 seconds to do so. Trying up to 300 required 7 seconds, checking through 344 million possibilities, and returned the above result and that result multiplied by 2.