Conference rusure::math

1604.0. "Euler's generalization of Fermat's Last Theorem" by TRACE::GILBERT (Ownership Obligates) Tue May 05 1992 21:51

    There was a generalization of Fermat's Last Theorem that held that
    
          n     n                 n
    (1)  a  +  b              =  x 	only for n <= 2;
    
          n     n     n           n
    (2)  a  +  b  +  c        =  x	only for n <= 3;
    
          n     n     n     n     n
    (3)  a  +  b  +  c  +  d  =  x	only for n <= 4, etc.
    
    However this was shown to be false for case (3) above...there is a set
    of 4 numbers whose 5th power adds up to an integral 5th power.  I have
    not heard of a counterexample to (2), so if you've got some spare CPU
    time and a clever algorithm, you can make history.
    
      John

[ John Hallyburton posted the above as note 1020.1 ]


Euler thought that Fermat's Last Theorem (which he first to prove for the
3-exponent case), could be generalized in this way.  In 1966, a computer
search found:

	27^5 + 84^5 + 110^5 + 133^5 = 144^5.

Recently it's been shown that there are infinitely many relatively prime
triples of 4th powers that sum to a 4th power.  The smallest such triple is

	95800^4 + 217519^4 + 414560^4 = 422481^4.

For 6th or greater powers, Euler's conjecture is still unresolved.


Reference:
"Old and New Unsolved Problems in Plane Geometry and Number Theory",
by Victor Klee and Stan Wagon.