Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How do you show that the equation $x^3+y^3+z^3=1$ has infinitely many solutions in integers? How about $x^3+y^3+z^3=2$?

share|cite|improve this question
up vote 13 down vote accepted

You can reduce the first equation to $$x^3 = -y^3, z = 1$$ with obvious infinite solutions. This paper details other families of solutions.

The second equation has solutions $(x,y,z)\equiv (6t^3+1, 1-6t^3, -6t^2)$ which (AFAIK) you find by construction (i.e you have to guess it).

share|cite|improve this answer
Yes, the solution you mentioned for the second one is exactly the same as the one I knew. Is this the only form of the solutions? – Amir Hossein Apr 13 '11 at 10:42
@Amir: for $n=2$ yes; it is the only known form but there are solutions that do not belong to any form. – Eelvex Apr 13 '11 at 10:47
An example which is not part of a known family is $1214928^3+3480205^3-3528875^3 = 2$. See – Tito Piezas III Nov 25 '14 at 23:29

For $x^3+y^3+z^3=1$ it is trivial - an infinite family of solutions is $(1,n,-n)$, and permutations of that.

For $x^3+y^3+z^3=2$ I'm not so sure there are infinitely many solutions. Are you just hypothesizing this, or do you know it to be true?

share|cite|improve this answer
Yes, the second one has infinitely many solutions. – Amir Hossein Apr 12 '11 at 16:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.