Just the trivial part $ (n,a,b,c)=(2k,1,-1,0)$ works

$ a+b =-c$ $ (a+b)^{n}= (-c)^{n}$ We have the similar problem of knowing when $ a^{n}+b^{n}+(-1)^{n}(a+b)^{n}$ is prime. If $ n$ is odd we have $ a^{n}+b^{n}-(a+b)^{n}$. Knowing the factorization of $ x^{n}-y^{n}$ shows that $ a^{n}$ is a common factor of this and hence the number is not prime. I have not yet done the case for even $ n$ which appears harder Nice problem I think! EDIT: oops, i did not look at mszew's post. The case of even $ n$ is then trivial .