A generalisation of the Euclid-Mullin sequences

TL;DR

This work generalises Mullin-type prime sequences to primes in fixed residue classes by introducing Generalised Euclid–Mullin polynomials (GEM) and specializing to cyclotomic polynomials $\Phi_m(cx)$. It shows that, under the Extended Riemann Hypothesis (ERH) and unconditionally for certain moduli, the analogue of the second Euclid–Mullin sequence, GEM_2$(\,\Phi_m(cx);1,m)$, omits infinitely many primes with $p\equiv1\pmod{m}$; it also proves unconditional results ensuring at least one omitted prime for infinitely many $m$. A precise criterion is given for when $\Phi_m(cx)$ is GEM$(1,m)$, via the set $S(m)$, and the paper obtains unconditional results for small $m$ and conditional results in general (via GLH/ERH). The approach combines cyclotomic-analytic machinery (Dedekind zeta, Burgess bounds, Dirichlet hyperbola), higher-order character arguments, and Diophantine inputs (Siegel-type results and Pell-type arguments) to extend known $m=1$ and $m=2^k$ cases to broader residue classes. Overall, the results advance our understanding of explicit Euclid-type constructions in prescribed residue classes and link prime-occurrence in these sequences to deep analytic and algebraic number-theoretic tools.

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 $Φ_m(cx)$ 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 $m=2^k$.

Theorems & Definitions (37)

• Definition 1.1
• Proposition 1.2
• proof
• Conjecture 1.3: Generalised Euclid--Mullin conjecture
• Theorem 1.4
• Proposition 1.5
• proof
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8