Hi, my name is Samuel Brown, i am a student at Wreake Valley College in Leicester, England and i came across your site on an excite.co.uk search. My question is a general knowledge one really and it comes from a book i was reading by Ian Stewart called 'Game, Set and Math...Enigmas and Conundrums'. In it there was one equation posed by Stewart which read, "x5 + y5 + z5 = w5 I have no idea whether or not this is possible." I am aiming to attempt at finding a solution to this (where x,y,z and w are integers), by some method i have yet to fathom. So my question to you is, "Do you know whether a solution(s) to this has been found? and if so what is it/are they? If not then do you know of any attempts at this equation?" I am most thankful for you assistance in my little project.

Hi Samuel,

Because of the analogy with Fermat's last theorem, it was thought at some point that at least n n-th powers were needed to sum up to a n-th power. If this were true, then x5 + y5 + z5 = w5 would have no solution. It is now known that this general conjecture is false, but the counterexample are quite recent.

The following information comes from Eric T. Bell, The Last Problem, 1990 edition, page 312.

The first counterexample to the conjecture was for n = 5 and appeared in Lander, L.J. and T.R. Parkin, Bulletin of the American mathematical Society 72 (1966) 84-92.

275 + 845 + 1105 + 1335 = 1445

Elkins, Noam D., On A4 + A4 + B4 + C4 = D4, Mathematics of Computation 51 (1988) #184 825-835, found that there are infinitely many solutions for n = 4, with the smallest being

958004 + 2175194 + 4145604 = 4224814

I don't know the status of x5 + y5 + z5 = w5.

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