# For which positive integers $n>2$ has the equation $a^n+b^n=2c^n$ non-trivial solutions?

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.

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 $2$.
• 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 ? – Peter Mar 23 '17 at 23:46