I just thought of it. I don't know if there is one such conjecture or a proven problem or if this is a new conjecture. If it already exists where can I find this problem. And most importantly how to prove it??
2026-03-27 15:36:16.1774625776
$P^2 + P + 1$ is prime if $P$ is prime.
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Set $P = 11$
$$P^2 + P + 1 = 121 + 11 + 1 = 133$$
$133$ is not prime, conjecture is not valid for all primes $P$.