(n+1, n^4+n^3+n^2+n+1)=1 Therefore if (n+1)(n^4+n^3+n^2+n+1) is a perfect square, then n^4+n^3+n^2+n+1 is a perfect square Let 4n^4+4n^3+4n^2+4n+4=k^2 (2n^2+n)^2=4n^4+4n^3+n^2 (2n^2+n+1)^2=4n^2+4n^3+5n^2+2n+1 When 5n^2+2n+1>=4n^2+4n+4 n^2-2n-3>=0 (n-3)(n+1)>=0 n>=3 or n3, 2n^2+n