Euler conjectured in 1769 that a^4 + b^4 + c^4 = d^4 has no solutions in positive integers.

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Problem

Euler conjectured in 1769 that a^4 + b^4 + c^4 = d^4 has no solutions in
positive integers.  Is he right?

[ Editor: problem statement reconstructed from the posting. ]

Show solution

Contrary to Euler's 1769 conjecture that the above equation has no
solutions in positive integers, I proved that there are infinitely
many, enough for the corresponding rational points
(x,y,z)=(+-a/d,+-b/d,+-c/d) on the surface x^4+y^4+z^4=1 to be dense
in the set of real solutions.  The smallest solution I know is

         4           4           4           4
  2682440  + 15365639  + 18796760  = 20615673  ;

the next has several dozen digits in each of a, b, c, d.

-- Noam D. Elkies (sci.math, August 1987)

[ Editor: posted under the subject "a^4+b^4+c^4=d^4"; the copy in the
collected material was reposted by Maarten Litmaath <maart@cs.vu.nl>. ]