A generalisation of the Euclid-Mullin sequences

We extend Mullin’s prime-generating procedures to produce sequences of primes lying in given residue classes. In particular we study the sequences generated by cyclotomic polynomials $Φ_m(cx)$ for suitable $c\in\mathbb{Z}$. Under the Extended Riemann Hypothesis in general and unconditionally for some moduli, we show that the analogue of the second Euclid–Mullin sequence omits infinitely many primes $\equiv1\pmod{m}$. We further show unconditionally that at least one prime is omitted for infinitely many $m$. This generalises work of the first author for $m=1$ and the second author for $m=2^k$.

💡 Research Summary

The paper revisits the classic Euclid‑Mullin prime‑generating procedures and places them in a much broader framework. Mullin’s original construction produced two infinite sequences of primes: the “small‑prime” sequence (at each step one takes the smallest prime divisor of the product of all previous terms plus one) and the “large‑prime” sequence (the analogous construction with the largest prime divisor). While many basic questions about the first sequence remain open, the second sequence is known to miss all primes below 47 and, as shown by the first author in earlier work, to omit infinitely many primes under mild hypotheses.

To study primes restricted to a fixed residue class a (mod m), the authors introduce the notion of a Generalised Euclid–Mullin polynomial, denoted GEM(a,m). A polynomial f∈ℤ