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

Question

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.

Answer 1

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 $2$.