Prove or disprove that ${F_{n}}^2 + 41$ is always a composite (if $F_{n}$ is $n^{th}$ Fibonacci number) conjectures that when $F$ is a fibonacci number, $F^2+41$ is composite. As reported there, computer calculations find that the conjecture is false; the smallest counterexample occurs at the $12588$th fibonacci number, which has 2630 digits.

Prove or disprove that ${F_{n}}^2 + 41$ is always a composite (if $F_{n}$ is $n^{th}$ Fibonacci number) conjectures that when $F$ is a fibonacci number, $F^2+41$ is composite. As reported there, computer calculations find that the conjecture is false; the smallest counterexample occurs at the $12588$th fibonacci number, which has 2630 digits.