Consider sequence $x^2-x+1$ ($1,3,7,13,21,31,43,57,73,91,\dots$). Let's consider prime factorization of each term. $$3=3$$ $$7=7$$ $$13=13$$ $$21=3\times7$$
It seems that the only prime factors we ever get are 3 and those of the form $6k+1$. In fact, prime factorization of the first 10 000 terms of the sequence gives 7233 distinct primes and all of them (except 3) are $6k+1$.
That no member of the sequence is ever divisible by a prime of the form $6k-1$ is a purely empirical conjecture. Is there a formal proof for it (or a counterexample)?