From: elkies@ramanujan.harvard.edu (Noam Elkies)
Newsgroups: sci.math,rec.puzzles
Subject: (x+y+z)^3=xyz [Re: Math Problems Site]
Date: 13 Aug 1995 23:16:28 GMT
Summary: No integer solutions with xyz nonzero.

In article <DD9EH0.AMB@agc.bio.ns.ca> ruxton@agc.bio.ns.ca
(Mike Ruxton (CHS)) writes:
>>>: Unsolved Problem 32:

>>>:   Can you find three positive integers x, y, and z,
>>>: such that (x+y+z)^3=xyz?  [impossible since (x+y+z)^3>=27xyz]

>What about if the integers aren't necessarily positive?

Obviously there are the solutions (x,-x,0), (0,y,-y), (-z,0,z) for
any integer x,y,z.  Less obviously, there are no other solutions.
This is an elliptic curve; converting to minimal Weierstrass form
on GP reveals that it is the curve Y^2+XY+Y=X^3 of conductor 26.
From either Tingley's tables in LNM 476, or Cremona's more recent
and extensive tables (where this curve is listed as 26-A and 26-A3
respectively) we learn that the above 3-torsion points are the
only rational points on the curve, and that this can be verified
by a 3-descent.

--Noam D. Elkies (elkies@math.harvard.edu)
  Dept. of Mathematics, Harvard University