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On the factorization of xdr + xdr−1 + ⋯ + xd2 + xd1 + p and a conjecture of C. Nicol

  • Michael Filaseta
  • Lilit Martirosyan
  • London Cameron Swan
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Abstract

For an arbitrary prime p and arbitrary positive integers r and d1, …, dr with dr > ⋯ > d1, we show that xdr + xdr−1 + ⋯ + xd2 + xd1 + p removed of all of its monic irreducible reciprocal factors is irreducible. As a consequence, we establish a number of results related to a conjecture of Charles Nicol that, over the rationals, the sum of two cyclotomic polynomials Φn(x) + Φm(x) is a product of cyclotomic polynomials times either $2$ or an irreducible polynomial, where n and m are integers exceeding $1$. We also establish similar results for Φn(x)Φm(x)+1.

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Cite

BibTeXRIS
BibTeX
@misc{filaseta2026factorization,
  title = {{On the factorization of $x^d_r + x^d_r-1 + \cdots + x^d_2 + x^d_1 + p$ and a conjecture of C. Nicol}},
  author = {Michael Filaseta and Lilit Martirosyan and London Cameron Swan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.05568},
  eprint = {2610.05568},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.05568},
}
RIS
TY  - UNPB
TI  - On the factorization of x^d_r + x^d_r−1 + ⋯ + x^d_2 + x^d_1 + p and a conjecture of C. Nicol
AU  - Michael Filaseta
AU  - Lilit Martirosyan
AU  - London Cameron Swan
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.05568
UR  - https://arxiv.org/abs/2610.05568
AB  - For an arbitrary prime p and arbitrary positive integers r and d_1, …, d_r with d_r > ⋯ > d_1, we show that x^d_r + x^d_r−1 + ⋯ + x^d_2 + x^d_1 + p removed of all of its monic irreducible reciprocal factors is irreducible. As a consequence, we establish a number of results related to a conjecture of Charles Nicol that, over the rationals, the sum of two cyclotomic polynomials Φ_n(x) + Φ_m(x) is a product of cyclotomic polynomials times either $2$ or an irreducible polynomial, where n and m are integers exceeding $1$. We also establish similar results for Φ_n(x)Φ_m(x)+1.
ER  -