@article{guy_AMM,
 ISSN = {00029890, 19300972},
 URL = {http://www.jstor.org/stable/2324367},
 author = {Richard Kenneth Guy},
 journal = {The American Mathematical Monthly},
 number = {2},
 pages = {169--172},
 publisher = {[Taylor & Francis, Ltd., Mathematical Association of America]},
 title = {Every Number is Expressible as the Sum of How Many Polygonal Numbers?},
 urldate = {2025-09-07},
 volume = {101},
 year = {1994}
}

@article{vinogradov,
 author = {I. M. Vinogradov},
 journal = {Dokl. Akad. Nauk. SSSR},
 number = {15},
 pages = {291--294},
 title = {Representation of an odd number as the sum of three primes},
 year = {1937}
}

@misc{helfgott,
    title={The ternary {G}oldbach problem},
    author={Harald Andres Helfgott},
    year={2015},
    eprint={1501.05438},
    archivePrefix={arXiv},
    primaryClass={math.NT},
    note={\url{https://arxiv.org/abs/1501.05438}}
}

@book{iwaniec_kowalski,
    author = {Henryk Iwaniec and Emmanuel Kowalski},
    title = {Analytic {N}umber {T}heory},
    publisher = {Colloquium Publications},
    year = {2004},
    volume = {53}
}

@MISC {stackexchange1,
    TITLE = {Solve $x^3+x \equiv 1 \pmod p$},
    AUTHOR = {lsr314 (https://math.stackexchange.com/users/66964/lsr314)},
    HOWPUBLISHED = {Mathematics Stack Exchange},
    NOTE = {URL: \url{https://math.stackexchange.com/q/378331} (version: 2013-05-01)},
    EPRINT = {https://math.stackexchange.com/q/378331},
    URL = {https://math.stackexchange.com/q/378331}
}

@MISC {stackexchange2,
    TITLE = {Understanding a result of Serre about zeros of $x^3 - x - 1$ in $\mathbb{F}_p$},
    AUTHOR = {Ethan Alwaise (https://math.stackexchange.com/users/221420/ethan-alwaise)},
    HOWPUBLISHED = {Mathematics Stack Exchange},
    NOTE = {URL: \url{https://math.stackexchange.com/q/2688853} (version: 2018-03-13)},
    EPRINT = {https://math.stackexchange.com/q/2688853},
    URL = {https://math.stackexchange.com/q/2688853}
}

@article{heath_brown_circle_method,
author = {David Rodney Heath-Brown},
year = {1996},
pages = {149--206},
title = {A new form of the circle method, and its application to quadratic forms},
journal = {Journal für die reine und angewandte Mathematik},
doi = {10.1515/crll.1996.481.149}
}

%month = {December},

@article{strongly_hardy_littlewood,
author = {Mikhail Borovoi and Ze\'ev Rudnick },
year = {1995},
month = {12},
pages = {37-66},
title = {Hardy-{L}ittlewood varieties and semisimple groups},
volume = {119},
journal = {Inventiones mathematicae}
}

@article{sum_of_three_squares_weird_logarithmic_factor,
author = {Duke, William and Rudnick, Zeev and Sarnak, Peter},
year = {1993},
month = {07},
title = {Density of integer points on affine homogeneous varieties},
volume = {71},
journal = {Duke Mathematical Journal},
doi = {10.1215/S0012-7094-93-07107-4}
}

@article{hardy_littlewood_def,
author = {Wolfgang M. Schmidt},
year = {1985},
month = {09},
title = {The density of integer points on homogeneous varieties},
volume = {154},
journal = {Acta Mathematica},
pages = {243-296}
}

@article{nathanson_cauchy,
 ISSN = {00029939, 10886826},
 URL = {http://www.jstor.org/stable/2046263},
 abstract = {This paper presents a simple proof that every nonnegative integer is the sum of m + 2 polygonal numbers of order m + 2.},
 author = {Melvyn B. Nathanson},
 journal = {Proceedings of the American Mathematical Society},
 number = {1},
 pages = {22--24},
 publisher = {American Mathematical Society},
 title = {A {S}hort {P}roof of {C}auchy's {P}olygonal {N}umber {T}heorem},
 urldate = {2025-10-14},
 volume = {99},
 year = {1987}
}

@article{toth_mod_4,
author = {T\'oth, L\'aszl\'o},
year = {2014},
volume = {17},
title = {Counting solutions of quadratic congruences in several variables revisited},
journal = {Journal of Integer Sequences},
note = {Article 14.11.6}
}

@article{bringmann,
author = {Kathrin Bringmann and Min-Joo Jang and Ben Kane and Alvin Cheuk Hin Tse},
year = {2024},
title = {Representations of integers as sums of four polygonal numbers and partial theta functions},
volume = {11},
journal = {Research in the Mathematical Sciences},
note = {Article Number 71}
}

@incollection{heath_brown_congruence_restriction,
  author    = {David Rodney Heath-Brown},
  title     = {Analytic {M}ethods for the {D}istribution of {R}ational {P}oints on {A}lgebraic {V}arieties},
  booktitle = {Equidistribution in Number Theory: An Introduction},
  editor    = {Granville, Andrew and Rudnick, Ze\'ev},
  series    = {NATO Science Series II: Mathematics, Physics and Chemistry},
  volume    = {237},
  pages     = {139--168},
  publisher = {Springer},
  address   = {Dordrecht},
  year      = {2007}
}

@article{hua,
    author = {Hua, Loo-Keng},
    title = {On {W}aring's Problem},
    journal = {The Quarterly Journal of Mathematics},
    volume = {9},
    number = {1},
    pages = {199-202},
    year = {1938},
    month = {01},
    issn = {0033-5606},
    doi = {10.1093/qmath/os-9.1.199},
    url = {https://doi.org/10.1093/qmath/os-9.1.199},
    eprint = {https://academic.oup.com/qjmath/article-pdf/os-9/1/199/4460503/os-9-1-199.pdf},
}

@article{Salie,
author = {Hans Salié},
journal = {Mathematische Zeitschrift},
pages = {91-109},
title = {Über die {K}loostermanschen {S}ummen {S}(u,v;q)},
url = {http://eudml.org/doc/168302},
volume = {34},
year = {1931--32},
}

@article{weil_bound,
author = {André Weil},
title = {On {S}ome {E}xponential {S}ums},
journal = {Proceedings of the National Academy of Sciences},
volume = {34},
number = {5},
pages = {204-207},
year = {1948},
doi = {10.1073/pnas.34.5.204},
URL = {https://www.pnas.org/doi/abs/10.1073/pnas.34.5.204},
eprint = {https://www.pnas.org/doi/pdf/10.1073/pnas.34.5.204}}

@article{hardy_squares_1918,
  title={On the {R}epresentation of a {N}umber as the Sum of any {N}umber of {S}quares, and in {P}articular of five or seven},
  author={Godfrey Harold Hardy},
  journal={Proceedings of the National Academy of Sciences of the United States of America},
  year={1918},
  volume={4 7},
  number={7},
  pages={189--93},
}

@article{kloosterman_refinement,
author = {Hendrik Douwe Kloosterman},
title = {{On the representation of numbers in the form $ax^2+by^2+cz^2+dt^2$}},
volume = {49},
journal = {Acta Mathematica},
number = {3-4},
publisher = {Institut Mittag-Leffler},
pages = {407--464},
year = {1926},
doi = {10.1007/BF02564120},
URL = {https://doi.org/10.1007/BF02564120}
}

@book{fermat,
    author = {Heath, Thomas L.},
    title = {Diophantus of {A}lexandria; a study in the history of {G}reek algebra},
    publisher = {Cambridge University Press},
    year = {1910},
    pages={188}
}

@article{cauchy,
author = {Augustin-Louis Cauchy},
title = {Démonstration du théorème général de Fermât sur les nombres polygones},
volume = {14},
journal = {Mém. Sei. Math. Phys. Inst. France},
pages = {177--220 = Oeuvres (2), vol. 6, 320-353.},
year = {1813--1815}
}

@book{berndt1998gauss,
  title={Gauss and {J}acobi {S}ums},
  author={Bruce C. Berndt and Ronald J. Evans and Kenneth S. Williams},
  isbn={9780471128076},
  lccn={97044440},
  series={Wiley-Interscience and Canadian Mathematics Series of Monographs and Texts},
  year={1998},
  publisher={Wiley}
}

@article{delta_method,
author = {William Duke and John Friedlander and Henryk Iwaniec},
journal = {Inventiones mathematicae},
number = {1},
pages = {1-8},
title = {Bounds for automorphic {$L$}-functions},
volume = {112},
year = {1993}
}

@book{legendre,
  title={Théorie des nombres},
  author={Adrien-Marie Legendre},
  edition={3rd},
  year={1830},
  volume={2},
  note={331--356}
}

@incollection{nathanson_legendre,
author="Nathanson, Melvyn B.",
editor="Adolphson, A. C.
and Conrey, J. B.
and Ghosh, A.
and Yager, R. I.",
title="Sums of {P}olygonal {N}umbers",
booktitle="Analytic Number Theory and Diophantine Problems: Proceedings of a Conference at Oklahoma State University, 1984",
year="1987",
publisher="Birkh{\"a}user Boston",
address="Boston, MA",
pages="305--316",
abstract="Let m ≥ 1. The k-th polygonal number of order m+2 is the sum of the first k terms of the arithmetic progression 1, 1+m, l+2m, l+3m,{\ldots} The polygonal numbers of orders 3 and 4 are the triangular numbers and squares, respectively.",
isbn="978-1-4612-4816-3",
doi="10.1007/978-1-4612-4816-3_17",
url="https://doi.org/10.1007/978-1-4612-4816-3_17"
}

@article{app1_of_delta_method,
author = {William Duke and John Friedlander and Henryk Iwaniec},
journal = {Inventiones mathematicae},
number = {2},
pages = {209--217},
title = {A quadratic divisor problem},
volume = {115},
year = {1994}
}

@article{app2_of_delta_method,
author = {William Duke and Henryk Iwaniec},
journal = {Compositio Mathematica},
number = {2},
pages = {145--158},
title = {Convolution {$L$}-series},
volume = {91},
year = {1994}
}


@book{gauss,
  title={Disquisitiones Arithmeticae},
  author={Carl Friedrich Gauss},
  edition={1st},
  year={1986},
  publisher={Springer New York, NY}
}

@article{lagrange,
author = {Joseph-Louis Lagrange},
journal = {Nouveaux mémoires de l’Académie royale des sciences et belles-lettres de Berlin, Année 1770},
pages = {123--133},
title = {Démonstration d’un théorème d’arithmétique},
year = {1772}
}

@article{supplement_to_legendre,
author = {Xiang-Zi Meng and Zhi-Wei Sun},
journal = {Acta Arithmetica},
pages = {229--249},
title = {Sums of four polygonal numbers with coefficients},
volume = {180},
year = {2017}
}

@article{k_8_universal,
title = {A result similar to {L}agrange's theorem},
journal = {Journal of Number Theory},
volume = {162},
pages = {190-211},
year = {2016},
issn = {0022-314X},
doi = {https://doi.org/10.1016/j.jnt.2015.10.014},
url = {https://www.sciencedirect.com/science/article/pii/S0022314X1500356X},
author = {Zhi-Wei Sun},
keywords = {Generalized octagonal numbers, Quadratic forms, Representations of integers},
abstract = {Generalized octagonal numbers are those p8(x)=x(3x−2) with x∈Z. In this paper we show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for 35 triples (b,c,d) with 1⩽b⩽c⩽d (including (2,3,4) and (2,4,8)), we prove that any nonnegative integer can be expressed as p8(w)+bp8(x)+cp8(y)+dp8(z) with w,x,y,z∈Z. We also pose several conjectures for further research.}
}

@article{k_minus_4_universal_k_geq_10,
title = {Fermat's polygonal number theorem for repeated generalized polygonal numbers},
journal = {Journal of Number Theory},
volume = {220},
pages = {163-181},
year = {2021},
author = {Soumyarup Banerjee and Manav Batavia and Ben Kane and Muratzhan Kyranbay and Dayoon Park and Sagnik Saha and Hiu Chun So and Piyush Varyani},}

@article{ternary_polygonal_haensch,
author={Anna Haensch and Ben Kane},
title={Almost universal ternary sums of polygonal numbers},
journal={Research in Number Theory},
year={2018},
month={Jan},
day={31},
volume={4},
note={article number 4},
}

@misc{finiteness_thm_kane,
      title={On finiteness theorems for sums of generalized polygonal numbers}, 
      author={Ben Kane and Zichen Yang},
      year={2023},
      eprint={2305.14670},
      archivePrefix={arXiv},
      primaryClass={math.NT},
      url={https://arxiv.org/abs/2305.14670},
      note={\url{https://arxiv.org/abs/2305.14670}}
}

@misc{prime_restriction_1,
      title={Sums of generalized polygonal numbers of almost prime ``length''}, 
      author={Soumyarup Banerjee and Ben Kane and Daejun Kim},
      year={2024},
      eprint={2409.07895},
      archivePrefix={arXiv},
      primaryClass={math.NT},
      url={https://arxiv.org/abs/2409.07895}, 
      note={\url{https://arxiv.org/abs/2409.07895}}
}

@misc{prime_restriction_2,
      title={Sums of four generalized polygonal numbers of almost prime length}, 
      author={Bosco Ng},
      year={2026},
      eprint={2603.04274},
      archivePrefix={arXiv},
      primaryClass={math.NT},
      url={https://arxiv.org/abs/2603.04274},
      note={\url{https://arxiv.org/abs/2603.04274}}
}

@misc{prime_restriction_3,
      title={Universal sums of generalized polygonal numbers of almost prime length}, 
      author={Soumyarup Banerjee and Ben Kane and Kwan To Ng},
      year={2026},
      eprint={2604.07826},
      archivePrefix={arXiv},
      primaryClass={math.NT},
      url={https://arxiv.org/abs/2604.07826},
      note={\url{https://arxiv.org/abs/2604.07826}}
}

@book{triangular_legendre,
  title={Trait{\'e} des fonctions elliptiques et des int{\'e}grales Euleriennes},
  author={Adrien-Marie Legendre},
  volume={3},
  year={1828},
}

@article{ono,
author={Ken Ono and Sinai Robins and Patrick T. Wahl},
title={On the representation of integers as sums of triangular numbers},
journal={aequationes mathematicae},
year={1995},
volume={50},
pages={73-94}
}

@article{exact_formulas_r_n_k_N,
title = {Sums of four polygonal numbers: {P}recise formulas},
journal = {Journal of Number Theory},
volume = {271},
pages = {407-422},
year = {2025},
issn = {0022-314X},
doi = {https://doi.org/10.1016/j.jnt.2025.01.016},
url = {https://www.sciencedirect.com/science/article/pii/S0022314X2500040X},
author = {Jialin Li and Haowu Wang},
keywords = {Polygonal numbers, Lagrange's four-square theorem, Jacobi forms, Fourier coefficients of automorphic forms, Representations of integers},
abstract = {In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized m-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are given as Z-linear combinations of Hurwitz class numbers. As applications, we prove several Zhi-Wei Sun's conjectures. As by-products, we obtain formulas for expressing the Fourier coefficients of ϑ(τ,z)4, η(τ)12, η(τ)4 and η(τ)8η(2τ)8 in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms.}
}

@book{jacobi_exact_formulas,
  title={Fundamenta nova theoriae functionum ellipticarum},
  author={Carl Gustav Jacob Jacobi},
  year={1829},
  publisher={Sumtibus Fratrum Borntraeger}
}

@article{liouville_12,
author={Joseph Liouville},
title={Extrait d'une lettre adressée à M. Besge},
journal={Journal de Mathématiques Pures et Appliquées},
year={1864},
volume={9},
series={2},
pages={296--298}
}

@article{liouville_10,
author = {Joseph Liouville},
journal = {Journal de Mathématiques Pures et Appliquées},
pages = {1--8},
title = {Nombre des représentations d'un entier quelconque sous la forme d'une somme de dix carrés.},
volume={11},
series={2},
year = {1866},
}

@article{waring_hardy_littlewood,
author = {Godfrey Harold Hardy and John Edensor Littlewood},
journal = {Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse},
pages = {33--54},
title = {Some problems of '{P}artitio numerorum'; {I}: {A} new solution of {W}aring's problem},
year = {1920},
}

@incollection{bateman,
author="Bateman, Paul T.",
editor="Berndt, Bruce C.
and Diamond, Harold G.
and Hildebrand, Adolf J.",
title="The asymptotic formula for the number of representations of an integer as a sum of five squares",
bookTitle="Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam Volume 1",
year="1996",
publisher="Birkh{\"a}user Boston",
address="Boston, MA",
pages="129--139",
abstract="In this paper we remark that the asymptotic formula for the number of representations of an integer as a sum of five squares can be derived by a simple elementary argument from Jacobi's formula for the number of representations of an integer as a sum of four squares. A by-product of our argument is an asymptotic formula for the sum {\$}{\$}{\backslash}sum{\backslash}limits{\_}{\{}|j| < n^{\{}{\{}1 {\backslash}mathord{\{}{\backslash}left/{\{}{\backslash}vphantom {\{}1 2{\}}{\}} {\backslash}right.{\backslash}kern-{\backslash}nulldelimiterspace{\}} 2{\}}{\}} {\}} {\{}{\backslash}sigma (n{\backslash}, - {\backslash},j^2 ),{\}}{\$}{\$}Where $\sigma$ denotes the sum-of-divisors function.",
isbn="978-1-4612-4086-0",
doi="10.1007/978-1-4612-4086-0_6",
url="https://doi.org/10.1007/978-1-4612-4086-0_6"
}

@article{heath_brown_cubic,
 ISSN = {1364503X, 14712962},
 URL = {http://www.jstor.org/stable/54887},
 abstract = {We use the Hardy-Littlewood circle method, in the form developed by Heath-Brown in 1996, to investigate the number of integer zeros of diagonal cubic forms. The results are subject to unproved hypotheses concerning certain Hasse-Weil L-functions. For six variables we show that there are O(P3+ε) zeros up to height P, for any ε > 0. For four variables we show that there are O(P3/2+ε) such zeros, excluding any that lie on rational lines in the corresponding surface.},
 author = {David Rodney Heath-Brown},
 journal = {Philosophical Transactions: Mathematical, Physical and Engineering Sciences},
 number = {1738},
 pages = {673--699},
 publisher = {Royal Society},
 title = {The Circle Method and Diagonal Cubic Forms},
 urldate = {2026-04-17},
 volume = {356},
 year = {1998}
}

@book{uspensky_heaslet,
  title={Elementary {N}umber {T}heory},
  author={James Uspensky and Maxwell Heaslet},
  year={1939},
  publisher={McGraw-Hill Book Company, Inc.}
}

@book{weil_book,
  title={Number Theory: An approach through history from {H}ammurapi to {L}egendre},
  author={André Weil},
  year={1984},
  publisher={Birkhäuser}
}

@book{Montgomery_Vaughan_2006, place={Cambridge}, series={Cambridge Studies in Advanced Mathematics}, title={Multiplicative {N}umber {T}heory {I}: {C}lassical {T}heory}, publisher={Cambridge University Press}, author={Montgomery, Hugh L. and Vaughan, Robert C.}, year={2006}, collection={Cambridge Studies in Advanced Mathematics}}

@article{convexity_davenport,
    author = {Davenport, H.},
    title = {On Dirichlet's L-Functions},
    journal = {Journal of the London Mathematical Society},
    volume = {s1-6},
    number = {3},
    pages = {198-202},
    year = {1931},
    month = {07},
    issn = {0024-6107},
    doi = {10.1112/jlms/s1-6.3.198},
    url = {https://doi.org/10.1112/jlms/s1-6.3.198},
    eprint = {https://academic.oup.com/jlms/article-pdf/s1-6/3/198/2861057/s1-6-3-198.pdf},
}

@book{hardy_wright,
  title={An {I}ntroduction to the {T}heory of {N}umbers},
  author={Godfrey Harold Hardy and Edward Maitland Wright},
  isbn={9780198531715},
  lccn={99461762},
  year={1979},
  publisher={Clarendon Press Oxford University Press},
  edition={5th}
}

@incollection{dirichlet_sums_of_squares,
author="Borwein, Jonathan Michael
and Choi, Kwok-Kwong Stephen",
editor="Berndt, Bruce
and Ono, Ken",
title="On {D}irichlet {S}eries for {S}ums of {S}quares",
bookTitle="Number Theory and Modular Forms: Papers in Memory of Robert A. Rankin",
year="2003",
publisher="Springer US",
address="Boston, MA",
pages="95--127"
}

@article{lattice_asymptotic,
 author = {Martin Widmer},
 journal = {Proceedings of the American Mathematical Society},
 number = {2},
 pages = {677--689},
 publisher = {American Mathematical Society},
 title = {Lipschitz class, narrow class, and counting lattice points},
 volume = {140},
 year = {2012}
}



















@Article{DSP,
author={Duke, William
and Schulze-Pillot, Rainer},
title={Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids},
journal={Inventiones mathematicae},
year={1990},
volume={99},
pages={49--57},
doi={10.1007/BF01234411},
url={https://doi.org/10.1007/BF01234411}
}


@Article{sun,
author={Dmitry Krachun and Zhi-Wei Sun},
title={On sums of four pentagonal numbers with coefficients},
journal={Electronic Research Archive},
year={2020},
volume={28},
number={1},
pages={559--566}
}

@article{bateman_three_squares,
 ISSN = {00029947, 10886850},
 URL = {http://www.jstor.org/stable/1990859},
 author = {Paul Trevier Bateman},
 journal = {Transactions of the American Mathematical Society},
 number = {1},
 pages = {70--101},
 publisher = {American Mathematical Society},
 title = {On the Representations of a Number as the Sum of Three Squares},
 urldate = {2026-08-25},
 volume = {71},
 year = {1951}
}

@article{siegel_mass_formula,
 ISSN = {0003486X, 19398980},
 URL = {http://www.jstor.org/stable/1968644},
 author = {Carl Ludwig Siegel},
 journal = {Annals of Mathematics},
 number = {3},
 pages = {527--606},
 publisher = {[Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University]},
 title = {Über Die Analytische Theorie Der Quadratischen Formen},
 urldate = {2026-08-25},
 volume = {36},
 year = {1935}
}

@article{van_der_blij_siegel_mass_formula,
 ISSN = {0003486X, 19398980},
 URL = {http://www.jstor.org/stable/1969584},
 author = {Frederik {van der} Blij},
 journal = {Annals of Mathematics},
 number = {4},
 pages = {875--883},
 publisher = {[Annals of Mathematics, Trustees of Princeton University on Behalf of the Annals of Mathematics, Mathematics Department, Princeton University]},
 title = {On the Theory of Quadratic Forms},
 urldate = {2026-08-25},
 volume = {50},
 year = {1949}
}

@article{robbins_gen_pentagonal_formula,
    author = {Neville Robbins},
    title = {On Sums of Two or Three Pentagonal Numbers},
    journal = {Ars Combinatoria},
    year = {2012},
    volume = {107},
    pages = {493--497}
}







@article{p_adic_density_comp,
author = {Jones, Edna},
title = {Local densities of diagonal integral ternary quadratic forms at odd primes},
journal = {International Journal of Number Theory},
volume = {17},
number = {3},
pages = {547-575},
year = {2021},
doi = {10.1142/S1793042120400357},
}


@misc{oeis_exceptional_set,
    Author = {{OEIS Foundation Inc. (2026)}},
    Note = {\url{https://oeis.org/A003679}},
    Title = {Numbers that are not the sum of 3 pentagonal numbers, {E}ntry {A}003679 in The {O}n-{L}ine {E}ncyclopedia of {I}nteger {S}equences}}

@book{last_notebook,
    AUTHOR = {Srinivasa Ramanujan},
     TITLE = {The lost notebook and other unpublished papers},
      NOTE = {Springer-Verlag, Reprinted 2008},
 PUBLISHER = {Narosa Publishing House, New Delhi; Berlin, New York},
      YEAR = {1988},
}

@article{AMDEBERHAN_ONO_SINGH_2025, 
title={Derivatives Of Theta Functions As Traces Of Partition {E}isenstein Series}, 
volume={258}, 
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