A generalisation of the Euclid-Mullin sequences

Abstract

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 $\hat{I}{¦}_m(cx)$ 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 $\equiv 1(\mathrm{mod}m)$. 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 $m=2^k$.

15 pages