I was trying to find counterexamples for the Euler conjecture for fifth power, but I couldn't find any aside from $27^5+84^5+110^5+133^5=144^5.$ Python script finds the first solution using a brute force approach in a second. It takes one day to check that there are no solutions below 1000 aside from multiples of the first solution.
In case of $a^4+b^4+c^4=d^4$ one can use modular arithmetic because $(n^4)\mod5$ guarantees that $a\mod5=0, b\mod 5=0, c\mod5=1$ and $d\mod5=1,$ allowing aggressive pruning. There are no such modulo for fifth power. One of the best cases $(x^5 \mod 11)$ yields [ 0 1 10 1 1 1 10 10 10 1 10], allowing to speed up the computation by a factor of $257,$ which isn't much.
What other tricks can be used to speed up the computation? Are there any other solutions known?
