0
$\begingroup$

For which positive integers $n>2$ has the equation $$a^n+b^n=2c^n$$ integer solutions with $0\le a<b$ ?

A small test with PARI/GP using this program :

? for(n=3,10,for(a=0,2000,for(b=a+1,2000,c=round(((a^n+b^n)/2)^(1/n));if(a^n+b^n
==2*c^n,print([a,b,c])))))
?

shows that for $3\le n\le 10$ and $0\le a<b\le 2000$, no solutions exist. So, they seem to be pretty rare at first glance.

$\endgroup$
1
$\begingroup$

All solutions are known (there are no primitive ones), see the article Winding quotients and some variants of Fermat’s Last Theorem by H. Darmon. equation $(1)$, Main Theorem on page $2$.

$\endgroup$
  • $\begingroup$ The article states that there are no non-trivial primitive solutions for $n\ge 3$. Does this imply that $a=b$ is the only way to get a solution ? $\endgroup$ – Peter Mar 23 '17 at 23:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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