@article{Nakayama1940,
  author = {Nakayama, Masayosi},
  title = {On the decomposition of a rational number into ``{Stammbr{\"u}che}''},
  journal = {Tohoku Mathematical Journal, First Series},
  volume = {46},
  year = {1940},
  pages = {1--21},
  url = {https://www.jstage.jst.go.jp/article/tmj1911/46/0/46_0_1/_article/-char/en},
  note = {\href{https://www.jstage.jst.go.jp/article/tmj1911/46/0/46_0_1/_pdf/-char/en}{Public text}}
}

@article{Erdos1950,
  author = {Erd{\H o}s, Paul},
  title = {Az {$1/x_1+1/x_2+\cdots+1/x_n=a/b$} egyenlet eg{\'e}sz sz{\'a}m{\'u} megold{\'a}sair{\'o}l},
  journal = {Matematikai Lapok},
  volume = {1},
  year = {1950},
  pages = {192--210},
  url = {https://users.renyi.hu/~p_erdos/1950-02.pdf},
  note = {\href{https://users.renyi.hu/~p_erdos/1950-02.pdf}{Public text}}
}

@book{ErdosGraham1980,
  author = {Erd{\H o}s, Paul and Graham, Ronald L.},
  title = {Old and New Problems and Results in Combinatorial Number Theory},
  series = {Monographies de L'Enseignement Math{\'e}matique},
  volume = {28},
  publisher = {L'Enseignement Math{\'e}matique},
  address = {Geneva},
  year = {1980},
  url = {https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf},
  note = {\href{https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf}{Public text}}
}

@article{Vose1985,
  author = {Vose, Michael D.},
  title = {{Egyptian} fractions},
  journal = {Bulletin of the London Mathematical Society},
  volume = {17},
  number = {1},
  year = {1985},
  pages = {21--24},
  doi = {10.1112/blms/17.1.21},
  note = {\href{https://doi.org/10.1112/blms/17.1.21}{Publisher version}}
}

@article{Takenouchi1921,
  author = {Takenouchi, Tanz{\^o}},
  title = {On an indeterminate equation},
  journal = {Proceedings of the Physico-Mathematical Society of Japan, Third Series},
  volume = {3},
  number = {6},
  year = {1921},
  pages = {78--92},
  doi = {10.11429/ppmsj1919.3.6_78},
  note = {\href{https://doi.org/10.11429/ppmsj1919.3.6_78}{Publisher version}}
}

@article{ErdosTuran1948,
  author = {Erd{\H o}s, Paul and Tur{\'a}n, P{\'a}l},
  title = {On a problem in the theory of uniform distribution. {I, II}},
  journal = {Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen},
  volume = {51},
  year = {1948},
  pages = {1146--1154, 1262--1269},
  url = {https://www.renyi.hu/~p_erdos/1948-02.pdf},
  note = {\href{https://www.renyi.hu/~p_erdos/1948-02.pdf}{Part I}; \href{https://users.renyi.hu/~p_erdos/1948-03.pdf}{Part II}}
}

@book{GrahamKolesnik1991,
  author = {Graham, S. W. and Kolesnik, G.},
  title = {Van der {Corput}'s Method of Exponential Sums},
  series = {London Mathematical Society Lecture Note Series},
  volume = {126},
  publisher = {Cambridge University Press},
  address = {Cambridge},
  year = {1991},
  doi = {10.1017/CBO9780511661976},
  note = {\href{https://doi.org/10.1017/CBO9780511661976}{Publisher version}}
}

@article{Selberg1949,
  author = {Selberg, Atle},
  title = {An elementary proof of the prime-number theorem},
  journal = {Annals of Mathematics. Second Series},
  volume = {50},
  number = {2},
  year = {1949},
  pages = {305--313},
  doi = {10.2307/1969455},
  note = {\href{https://doi.org/10.2307/1969455}{Publisher version}}
}

@misc{Bloom304,
  author = {Bloom, Thomas F.},
  title = {{Erd{\H o}s Problem 304}},
  howpublished = {\url{https://www.erdosproblems.com/304}},
  note = {Accessed 21 September 2026}
}

@article{vanDoornTang2026,
  author = {van Doorn, Wouter and Tang, Quanyu},
  title = {The smallest denominator not contained in a unit fraction decomposition of {$1$} with fixed length},
  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
  year = {2026},
  pages = {1--9},
  doi = {10.1017/S0305004126102102},
  note = {Published online 8 July 2026; arXiv:2512.22083v2; \href{https://doi.org/10.1017/S0305004126102102}{Publisher version}},
  url = {https://arxiv.org/abs/2512.22083v2}
}

@article{Konyagin2014,
  author = {Konyagin, S. V.},
  title = {Double exponential lower bound for the number of representations of unity by {Egyptian} fractions},
  journal = {Matematicheskie Zametki},
  volume = {95},
  number = {2},
  year = {2014},
  pages = {312--316},
  doi = {10.4213/mzm10417},
  url = {https://www.mathnet.ru/eng/mzm10417},
  note = {In Russian; \href{https://doi.org/10.4213/mzm10417}{Publisher version}}
}

@article{Elsholtz2016,
  author = {Elsholtz, Christian},
  title = {{Egyptian} fractions with odd denominators},
  journal = {The Quarterly Journal of Mathematics},
  volume = {67},
  number = {3},
  year = {2016},
  pages = {425--430},
  doi = {10.1093/qmath/haw020},
  url = {https://arxiv.org/abs/1606.02117},
  note = {\href{https://doi.org/10.1093/qmath/haw020}{Publisher version}}
}

@article{ElsholtzPlanitzer2021,
  author = {Elsholtz, Christian and Planitzer, Stefan},
  title = {Sums of four and more unit fractions and approximate parametrizations},
  journal = {Bulletin of the London Mathematical Society},
  volume = {53},
  number = {3},
  year = {2021},
  pages = {695--709},
  doi = {10.1112/blms.12452},
  url = {https://arxiv.org/abs/2012.05984},
  note = {\href{https://doi.org/10.1112/blms.12452}{Publisher version}}
}


@article{Kumchev1997,
  author = {Kumchev, Angel},
  title = {On the {Piatetski-Shapiro-Vinogradov} theorem},
  journal = {Journal de Th{\'e}orie des Nombres de Bordeaux},
  volume = {9},
  number = {1},
  year = {1997},
  pages = {11--23},
  doi = {10.5802/jtnb.186},
  url = {https://jtnb.centre-mersenne.org/item/JTNB_1997__9_1_11_0/},
  note = {\href{https://doi.org/10.5802/jtnb.186}{Publisher version}}
}

@misc{Bloom293,
  author = {Bloom, Thomas F.},
  title = {{Erd{\H o}s Problem 293}},
  howpublished = {\url{https://www.erdosproblems.com/293}}
}

@article{TenenbaumYokota1990,
  author = {Tenenbaum, G{\'e}rald and Yokota, Hisashi},
  title = {Length and denominators of {Egyptian} fractions, {III}},
  journal = {Journal of Number Theory},
  volume = {35},
  number = {2},
  year = {1990},
  pages = {150--156},
  doi = {10.1016/0022-314X(90)90109-5},
  url = {https://tenenb.perso.math.cnrs.fr/PPP/Egyptian.pdf},
  note = {\href{https://doi.org/10.1016/0022-314X(90)90109-5}{Publisher version}}
}

@article{Erdos1952Divisors,
 author={Erd{\H{o}}s, Paul},
 title={On the sum {$\sum_{k=1}^{x}d(f(k))$}},
 journal={Journal of the London Mathematical Society},
 volume={27}, number={1}, year={1952}, pages={7--15},
 doi={10.1112/jlms/s1-27.1.7},
 url={https://www.renyi.hu/~p_erdos/1952-08.pdf},
  note = {\href{https://doi.org/10.1112/jlms/s1-27.1.7}{Publisher version}}
}

@article{HildebrandTenenbaum1993,
 author={Hildebrand, Adolf and Tenenbaum, G{\'e}rald},
 title={Integers without large prime factors},
 journal={Journal de Th{\'e}orie des Nombres de Bordeaux},
 volume={5}, number={2}, year={1993}, pages={411--484},
 url={https://www.numdam.org/item/JTNB_1993__5_2_411_0/},
  note = {\href{https://www.numdam.org/item/JTNB_1993__5_2_411_0/}{Public text}}
}

@article{Gallagher1971,
 author={Gallagher, P. X.},
 title={A larger sieve},
 journal={Acta Arithmetica}, volume={18}, number={1},
 year={1971}, pages={77--81}, doi={10.4064/aa-18-1-77-81},
 url={https://matwbn.icm.edu.pl/ksiazki/aa/aa18/aa1819.pdf},
  note = {\href{https://doi.org/10.4064/aa-18-1-77-81}{Publisher version}}
}

@article{KumchevTolev2005,
 author={Kumchev, Angel V. and Tolev, Dimitar I.},
 title={An invitation to additive prime number theory},
 journal={Serdica Mathematical Journal},
 volume={31}, year={2005}, pages={1--74},
 url={https://tigerweb.towson.edu/akumchev/a17.pdf},
  note = {\href{https://tigerweb.towson.edu/akumchev/a17.pdf}{Author version}}
}

@article{Martin2000,
 author={Martin, Greg},
 title={Denser {Egyptian} fractions},
 journal={Acta Arithmetica},
 volume={95}, number={3}, year={2000}, pages={231--260},
 url={https://arxiv.org/abs/math/9811112},
  note = {\href{https://arxiv.org/abs/math/9811112}{Author version}}
}

@article{Croot1999,
 author={Croot, III, Ernest S.},
 title={On some questions of {Erd{\H{o}}s} and {Graham} about {Egyptian} fractions},
 journal={Mathematika},
 volume={46}, number={2}, year={1999}, pages={359--372},
 doi={10.1112/S0025579300007828},
 url={https://ecroot.math.gatech.edu/erdos_graham.pdf},
  note = {\href{https://doi.org/10.1112/S0025579300007828}{Publisher version}}
}

@incollection{GlibichukKonyagin2007,
 author={Glibichuk, A. A. and Konyagin, S. V.},
 title={Additive properties of product sets in fields of prime order},
 booktitle={Additive Combinatorics},
 series={CRM Proceedings and Lecture Notes},
 volume={43}, year={2007}, pages={279--286},
 publisher={American Mathematical Society},
 url={https://arxiv.org/abs/math/0702729},
  note = {\href{https://arxiv.org/abs/math/0702729}{Author version}}
}