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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.21901 |
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| _version_ | 1866908804490199040 |
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| author | Booker, Andrew R. Simon, Omri |
| author_facet | Booker, Andrew R. Simon, Omri |
| contents | 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 $m=2^k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21901 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A generalisation of the Euclid-Mullin sequences Booker, Andrew R. Simon, Omri Number Theory 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 $m=2^k$. |
| title | A generalisation of the Euclid-Mullin sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2601.21901 |