Let $\ a,b,c\ $ be integers with $\ 0<a<b<c\ $ and let $\ m\ $ be the smallest non-negative integer such that $$a^m+b^m+c^m$$ is composite.
Define $$f(a,b,c)=m-1$$ Hence $\ f(a,b,c)\ $ is the largest non-negative integer $\ m\ $ such that $$a^k+b^k+c^k$$ is prime for $\ k=0,\cdots ,m\ $. Since for $\ k=0\ $, we get always $\ 3\ $ which is a prime, $\ f(a,b,c)\ $ is always non-negative.
Has $f(a,b,c)$ a maximum ?
I have found $$f(2, 186, 803)=8$$ with brute force, that means that $\ 2^k+186^k+803^k\ $ is prime for $\ k=0,1,2,3,4,5,6,7,8\ $ which is unbeaten , if $\ c\le 1000\ $ holds.