Generalised Euclid–Mullin conjecture

Let $a,m\in\mathbb{N}$ satisfy

$$a^2\equiv1\pmod{m}.$$

Let $\operatorname{GEM}(a,m)$ be the class of polynomials introduced in the paper, and let $\operatorname{GEM}_1(f;a,m)$ and $\operatorname{GEM}_2(f;a,m)$ denote the first and second generalised Euclid–Mullin sequences associated to $f\in\operatorname{GEM}(a,m)$, obtained by choosing the smallest and largest eligible prime, respectively. Generalised Euclid–Mullin conjecture. (i) There exists $f\in\operatorname{GEM}(a,m)$ such that $\operatorname{GEM}_1(f;a,m)$ contains every prime $p\equiv a\pmod{m}$. (ii) For every $f\in\operatorname{GEM}(a,m)$, $\operatorname{GEM}_2(f;a,m)$ omits infinitely many primes $p\equiv a\pmod{m}$. The conjecture generalises the expected behaviour of Mullin's original sequences in prescribed arithmetic progressions. The supplied text does not establish either assertion or provide a resolution status beyond presenting them as conjectures.