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arXiv · 2601.21901

A generalisation of the Euclid-Mullin sequences

Andrew R. Booker·Omri Simon
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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 $\Phi_m(cx)$ 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 $m=2^k$.

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Andrew R. Booker, Omri Simon. 2026-01-29. A generalisation of the Euclid-Mullin sequences. https://arxiv.org/abs/2601.21901

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