has no solutions. This is Euler's conjecture.
This was proved many years later by a counter example by another mathematician.
(I am sorry, I don't remember the numbers in the counter example. I shall make a note of it and post later)
The counter example had numbers all greater than a million. Later on, it was also proved that any number of solutions exist in the realm of numbers after that.
Where had the one eyed cyclops, the most gifted mathematician the planet had ever seen, gone wrong?
If I remember right, he had tested numbers only upto a million. Mathematics had shown us yet again that if something is true for every number up to a million, it is not necessarily true thereafter too.
(I had read this in the book FLT by Simon Singh.)
It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
I have found
95800[sup]4[/sup] + 217519[sup]4[/sup] + 414560[sup]4[/sup] = 422481[sup]4[/sup]
But nothing involving powers of 3
Euler also conjectured that Orthogonal Latin Squares don't exist for 4n + 6 for every n. This is false, and while the counter example is not in the size of the millions, latin squares get very difficult to find for any number greater than 5 or so.