Pith. sign in

REVIEW 1 major objections 119 references

Gross-Zagier formula for the $4, 7$ cases of Sylvester's conjecture

T0 review · 1 major / 0 minor · reviewed 2026-07-03 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Explicit Gross-Zagier formula proven relating CM point heights to L-function derivatives for E_{p^i}

desk verdict Yin states an explicit Gross-Zagier height formula for the CM points built in the prior paper on the 4 and 7 mod 9 cases of Sylvester's conjecture. read the letter →

arxiv 2607.01744 v1 pith:DGFKITOT submitted 2026-07-02 math.NT

Hongbo Yin This is my paper
classification math.NT
keywords Gross-ZagierformulaCMpointsSylvester'sconjectureellipticcurvesL-functionsHeegnercanonicalheight
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The author had previously constructed CM points on the elliptic curves E_{p^i} given by y squared equals x cubed plus p to the 2i over 4 for primes p congruent to 4 or 7 modulo 9 and i equal to 1 or 2; these points produce rational solutions to x cubed plus y cubed equals p to the i and thereby settle the 4 and 7 cases of Sylvester's conjecture. This paper proves the explicit Gross-Zagier formula that equates the canonical height of these CM points to a constant multiple of the derivative at s equals 1 of the L-function attached to E_{p^i}. A sympathetic reader cares because the formula supplies a direct, computable bridge between the arithmetic size of the constructed points and the analytic behavior of the L-functions. If the relation holds, the non-torsion rational points on the cubic curves have heights that are determined by the vanishing order and derivative of the corresponding L-functions.

What carries the argument

The explicit Gross-Zagier formula that equates the Néron-Tate height of the constructed CM points on E_{p^i} to a multiple of L'(E_{p^i},1)

What would settle it

Direct numerical computation of the canonical height of the CM point for p=7 together with the value of L'(E_7,1); a mismatch beyond the predicted constant factor would falsify the formula.

Watch

Extended reading notes

Core claim

We prove the explicit Gross-Zagier formula relating the height of our CM points and the derivative of the L-functions of E_{p^i}.

Load-bearing premise

The CM points constructed in the prior work satisfy the Heegner point conditions needed for the explicit Gross-Zagier formula to apply.

Editorial extensions

If this is right

  • The canonical heights of the CM points are given explicitly by the L-function derivatives.
  • The rational points on x cubed plus y cubed equals p to the i have heights determined by analytic data.
  • The formula supplies a verification of the expected height-L-function relation in these two families of curves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Numerical checks of the formula for the smallest primes p=7 and p=13 become feasible once both sides are computed independently.
  • The same relation could be tested on any future CM points constructed for higher exponents or other congruence classes.
  • The formula links the geometric construction of rational points on the cubic surfaces directly to the analytic rank of the associated elliptic curves.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove an explicit Gross-Zagier formula equating the canonical height of CM points on the elliptic curves E_{p^i}: y^2 = x^3 + p^{2i}/4 (p prime ≡4 or 7 mod 9, i=1,2), constructed in the author's prior work [Yin26], to a multiple of L'(E_{p^i},1). These points are asserted to yield rational points on x^3 + y^3 = p^i, addressing the 4,7 cases of Sylvester's conjecture.

Significance. If the CM points satisfy the Heegner conditions on the appropriate modular curve, the explicit formula would furnish a concrete arithmetic-analytic relation for these specific curves, potentially confirming non-torsion points or providing evidence toward BSD in the context of the cubic Diophantine equations. The work supplies a targeted instance of Gross-Zagier but inherits its strength from the validity of the prior construction.

major comments (1)
  1. The central claim applies the explicit Gross-Zagier formula, which requires the constructed points to be Heegner points on X_0(N) (N the conductor of E_{p^i}) with matching discriminant and Atkin-Lehner eigenvalue conditions. No independent verification or explicit computation confirming these modular-curve conditions for p ≡ 4,7 mod 9 appears in the manuscript; the argument rests entirely on the construction in [Yin26] without additional checks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript. We respond to the single major comment below.

read point-by-point responses

  1. Referee: The central claim applies the explicit Gross-Zagier formula, which requires the constructed points to be Heegner points on X_0(N) (N the conductor of E_{p^i}) with matching discriminant and Atkin-Lehner eigenvalue conditions. No independent verification or explicit computation confirming these modular-curve conditions for p ≡ 4,7 mod 9 appears in the manuscript; the argument rests entirely on the construction in [Yin26] without additional checks.

    Authors: The CM points in question are constructed in [Yin26] precisely so that they are Heegner points on the appropriate X_0(N) satisfying the required discriminant and Atkin-Lehner conditions for each E_{p^i} with p ≡ 4 or 7 mod 9. The present paper takes those points as given and derives the explicit Gross-Zagier formula relating their heights to L'(E_{p^i},1). We will revise the introduction to add an explicit sentence stating that the Heegner hypotheses are verified in [Yin26] and therefore apply here, thereby making the dependence on the prior work more transparent without repeating the full construction. revision: partial

Circularity Check

1 steps flagged · score 6.0 of 10

Gross-Zagier application load-bearing on self-cited CM point construction without independent Heegner verification

  1. self citation load bearing [Abstract]
    "In \cite{Yin26}, the author constructed some CM points on the elliptic curves $E_{p^i}:y^2=x^3+\frac{p^{2i}}{4}$ for primes $p\equiv 4,7\mod 9$ and $i=1,2$, which give rational points on the curves $x^3+y^3=p^i$. This solves the $4,7$ cases of Sylvester's conjecture. In this paper, we prove the explicit Gross-Zagier formula relating the height of our CM points and the derivative of the $L$-functions of $E_{p^i}$."

    The explicit formula is asserted for these points, but their status as valid Heegner points on the modular curve X_0(N) (with matching conductor N of E_{p^i}, correct discriminant, and Atkin-Lehner eigenvalues) is justified solely by the overlapping-author citation to [Yin26]. No re-verification or explicit level/discriminant check appears in the present manuscript, so the central claim reduces to the prior self-construction.

full rationale

The paper's central result is an explicit Gross-Zagier formula equating heights of the constructed CM points to L'(E_{p^i},1). This requires the points to be Heegner points on X_0(N) satisfying the necessary discriminant, level, and Atkin-Lehner conditions for the curves E_{p^i}. The manuscript imports this status entirely via self-citation to the author's prior work [Yin26] and supplies no independent check or explicit matching of those modular conditions in the present text. While Gross-Zagier itself is an external theorem, the applicability to these specific points reduces to the self-cited construction, producing moderate circularity on the load-bearing premise.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract alone supplies no information on free parameters, axioms, or invented entities; all arrays are therefore empty.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gross-Zagier formula for the $4, 7$ cases of Sylvester's conjecture." pith.science (2026). https://pith.science/paper/DGFKITOT

@misc{pith2026260701744,
  author       = {Pith},
  title        = {Pith review of: Gross-Zagier formula for the $4, 7$ cases of Sylvester's conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGFKITOT}},
  note         = {Machine review of arXiv:2607.01744}
}
abstract

In \cite{Yin26}, the author constructed some CM points on the elliptic curves $E_{p^i}:y^2=x^3+\frac{p^{2i}}{4}$ for primes $p\equiv 4,7\mod 9$ and $i=1,2$, which give rational points on the curves $x^3+y^3=p^i$. This solves the $4,7$ cases of Sylvester's conjecture. In this paper, we prove the explicit Gross-Zagier formula relating the height of our CM points and the derivative of the $L$-functions of $E_{p^i}$.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

119 extracted references · 119 canonical work pages

  1. [1]

    Heegner , Date-Added =

    K. Heegner , Date-Added =. Diophantische analysis und modulfunktionen , Volume =. Math. Z. , Number =

    work page

  2. [2]

    Katz, N. M. , Date-Added =. p-adic ineterpolation of real analytic. Ann. of Math. , Number =

    work page

  3. [3]

    , Date-Added =

    Lieman, D. , Date-Added =. Nonvanishing of. Ann. of Math. , Pages =

    work page

  4. [4]

    Ribet, K. A. , Journal =. On modular representations of

    work page

  5. [5]

    Cox, D. A. , Date-Added =. Primes of the Form x^2+n y^2 , Year =

    work page

  6. [6]

    , Booktitle =

    Tate, J. , Booktitle =. Fourier analysis in number fields and

    work page

  7. [7]

    , Date-Added =

    Lang, S. , Date-Added =. Elliptic functions , Volume =

    work page

  8. [8]

    and Langlands, R

    Jacquet, H. and Langlands, R. P. , Date-Modified =. Automorphic Forms on GL(2): Part 1 , Url =. 1970 , Bdsk-Url-1 =

    work page 1970

Show all 119 references
  1. [9]

    Sylvester, J. J. , Journal =. On certain tenary cubic-form equations , Url =. 1879 , Bdsk-Url-1 =

  2. [10]

    Sylvester, J. J. , Date-Modified =. On certain tenary cubic-form equations , Url =. Amer. J. Math. , Language =. 1880 , Bdsk-Url-1 =

  3. [12]

    , Journal =

    Ligozat, G. , Journal =. Fonction L des courbes modulaires , Url =. 1969-1970 , Bdsk-Url-1 =

    1969

  4. [13]

    , Journal =

    Kobayashi, S. , Journal =. The p -adic

  5. [14]

    Gross, B. H. and Zagier, D. B. , Journal =. Heegner points and derivatives of. 1986 , Bdsk-Url-1 =

    1986

  6. [15]

    and Lehner, J

    Atkin, A.O.L. and Lehner, J. , Date-Modified =. Hecke Operators on. Mathematische Annalen , Keywords =. 1970 , Bdsk-Url-1 =

    1970

  7. [16]

    Kenku, M. A. and Momose, F. , Journal =. Automorphism groups of the modular curves. 1988 , Bdsk-Url-1 =

    1988

  8. [17]

    and Rogers, Nicholas F

    Elkies, Noam D. and Rogers, Nicholas F. , Booktitle =. Elliptic curves. 2004 , Bdsk-Url-1 =. doi:10.1007/978-3-540-24847-7_13 , Mrclass =

    2004 doi

  9. [18]

    Un analogue du calcul de

    Satg. Un analogue du calcul de. Invent. Math. , Mrclass =. 1987 , Bdsk-Url-1 =. doi:10.1007/BF01389425 , Fjournal =

    1987 doi

  10. [19]

    Liverance , Doi =

    E. Liverance , Doi =. A Formula for the Root Number of a Family of Elliptic Curves , Url =. Journal of Number Theory , Number =. 1995 , Bdsk-Url-1 =

    1995

  11. [20]

    Gross , Issn =

    Benedict H. Gross , Issn =. Local Orders, Root Numbers, and Modular Curves , Url =. American Journal of Mathematics , Number =. 1988 , Bdsk-Url-1 =

    1988

  12. [21]

    Yuan, Xinyi and Zhang, Shou-Wu and Zhang, Wei , Isbn =. The

  13. [22]

    and Voight, J

    Dasgupta, S. and Voight, J. , Journal =. Sylvester's problem and mock Heegner points , Volume =

  14. [23]

    Li,Yongxiong and Liu, Yu and Tian, Ye , Journal =. On the

  15. [24]

    and Shu, J

    Cai, L. and Shu, J. and Tian, Y. , Date-Modified =. Cube sum problem and an explicit. Amer. Jour. of Math. , Number =

  16. [25]

    Hu,Y. , Doi =. Cuspidal part of an. Res. Number Theory , Mrclass =. 2016 , Bdsk-Url-1 =

    2016

  17. [26]

    , Coden =

    Jedrzejak,T. , Coden =. Height estimates on cubic twists of the. Bull. Austral. Math. Soc. , Mrclass =. 2005 , Bdsk-Url-1 =. doi:10.1017/S0004972700034997 , Fjournal =

    2005 doi

  18. [27]

    Groupes de

    Satg. Groupes de. J. Number Theory , Mrclass =. 1986 , Bdsk-Url-1 =. doi:10.1016/0022-314X(86)90075-2 , Fjournal =

    1986 doi

  19. [28]

    , Journal =

    Milne, J.S. , Journal =

  20. [29]

    and Rapoport, M

    Deligne, P. and Rapoport, M. , Booktitle =. Les Sch\'emas de Modules de Courbes Elliptiques , Ty =. 1973 , Bdsk-Url-1 =. doi:10.1007/978-3-540-37855-6

    1973 doi

  21. [30]

    Selmer, E. S. , Journal =. The. 1951 , Bdsk-Url-1 =

    1951

  22. [31]

    Saito, Hiroshi , Journal =. On. 1993 , Bdsk-Url-1 =

    1993

  23. [32]

    On Some Results of

    Casselman, William , Journal =. On Some Results of. 1973 , Bdsk-Url-1 =

    1973

  24. [33]

    , Journal =

    Perrin-Riou, B. , Journal =. Points de Heegner et deriv. 1987 , Bdsk-Url-1 =

    1987

  25. [34]

    Kolyvagin, V. A. , Booktitle =. Euler systems , Volume =

  26. [35]

    Silverman, J. H. , Doi =. Advanced topics in the arithmetic of elliptic curves , Url =. 1994 , Bdsk-Url-1 =

    1994

  27. [36]

    Silverman, J. H. , Isbn =. The arithmetic of elliptic curves , Volume =

  28. [37]

    and Shu, J

    Cai, L. and Shu, J. and Tian, Y. , Doi =. Explicit. Algebra & Number Theory , Mrclass =. 2014 , Bdsk-Url-1 =

    2014

  29. [38]

    and Kramarz, G

    Zagier, D. and Kramarz, G. , Coden =. Numerical investigations related to the. J. Indian Math. Soc. (N.S.) , Mrclass =

  30. [39]

    Topics in classical automorphic forms , Url =

    Iwaniec, Henryk , Doi =. Topics in classical automorphic forms , Url =. 1997 , Bdsk-Url-1 =

    1997

  31. [40]

    , Isbn =

    Shimura, G. , Isbn =. Introduction to the arithmetic theory of automorphic functions , Volume =

  32. [41]

    and Voight, J

    Dasgupta, S. and Voight, J. , Journal =. Heegner points and Sylvester's conjecture , Volume =

  33. [42]

    Dickson, L. E. , Publisher =. History of the Theory of Numbers: Diophantine analysis , Volume =

  34. [43]

    and Viray, B

    Lauter, K. and Viray, B. , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2015 , NUMBER =. doi:10.1093/imrn/rnu223 , URL =

    2015 doi

  35. [44]

    Gross, B. H. and Zagier, D. B. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1985 , PAGES =

    1985

  36. [45]

    and Tate, J

    Serre, J-P. and Tate, J. , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1968 , PAGES =. doi:10.2307/1970722 , URL =

    1968 doi

  37. [46]

    GH from MO , Journal=

  38. [47]

    Wittmann, Christian , TITLE =. J. Number Theory , FJOURNAL =. 2001 , NUMBER =. doi:10.1006/jnth.2000.2622 , URL =

    2001 doi

  39. [48]

    and Shu, J

    Hu, Y. and Shu, J. and Yin, H. , TITLE =. Acta Arith. , FJOURNAL =. 2020 , NUMBER =. doi:10.4064/aa190212-3-10 , URL =

    2020 doi

  40. [49]

    Stephens, N. M. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1968 , PAGES =. doi:10.1515/crll.1968.231.121 , URL =

    1968 doi

  41. [50]

    Birch, B. J. and Stephens, N. M. , TITLE =. Topology , FJOURNAL =. 1966 , PAGES =. doi:10.1016/0040-9383(66)90021-8 , URL =

    1966 doi

  42. [51]

    and Shu, J

    Hu, Y. and Shu, J. and Yin, H. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBER =. doi:10.1090/tran/7760 , URL =

    2019 doi

  43. [52]

    , Journal=

    Zagier,D. , Journal=. Modular forms of one variable, Notes based on a course given in Utrecht, Spring 1991 , Year=

    1991

  44. [53]

    , TITLE =

    Tian, Y. , TITLE =. Camb. J. Math. , FJOURNAL =. 2014 , NUMBER =. doi:10.4310/CJM.2014.v2.n1.a4 , URL =

    2014 doi

  45. [54]

    Sagemath , Journal=. , Year=

  46. [55]

    , TITLE =

    Morain, F. , TITLE =. J. Th\'. 2007 , NUMBER =

    2007

  47. [56]

    Shu, Jie and Song, Xu and Yin, Hongbo , TITLE =. Math. Z. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00209-021-02730-w , URL =

    2021 doi

  48. [57]

    , TITLE =

    Tate, J. , TITLE =. Modular functions of one variable,. 1975 , MRCLASS =

    1975

  49. [58]

    , TITLE =

    Murabayashi, N. , TITLE =. J. Number Theory , FJOURNAL =. 2004 , NUMBER =. doi:10.1016/j.jnt.2004.05.004 , URL =

    2004 doi

  50. [59]

    Numbers of solutions of equations in finite fields , JOURNAL =

    Weil, Andr\'. Numbers of solutions of equations in finite fields , JOURNAL =. 1949 , PAGES =. doi:10.1090/S0002-9904-1949-09219-4 , URL =

    1949 doi

  51. [60]

    and Rosen, Michael I

    Ireland, Kenneth F. and Rosen, Michael I. , TITLE =. 1982 , PAGES =

    1982

  52. [61]

    Nagoya Math

    Shimura, Goro , TITLE =. Nagoya Math. J. , FJOURNAL =. 1971 , PAGES =

    1971

  53. [62]

    Algebraic number theory , Url =

    Neukirch, J\". Algebraic number theory , Url =. 1999 , Bdsk-Url-1 =. doi:10.1007/978-3-662-03983-0 , Isbn =

    1999 doi

  54. [63]

    Modular elliptic directions with complex multiplication (with an application to

    Gonz\'. Modular elliptic directions with complex multiplication (with an application to. Comment. Math. Helv. , FJOURNAL =. 2011 , NUMBER =. doi:10.4171/CMH/225 , URL =

    2011 doi

  55. [64]

    The solution of a base change problem for

    G\'. The solution of a base change problem for. Automorphic forms, representations and. 1979 , MRCLASS =

    1979

  56. [65]

    , TITLE =

    Gross, Benedict H. , TITLE =. 1980 , PAGES =

    1980

  57. [66]

    Atkin, A. O. L. and Li, Wen Ch'ing Winnie , TITLE =. Invent. Math. , FJOURNAL =. 1978 , NUMBER =. doi:10.1007/BF01390245 , URL =

    1978 doi

  58. [67]

    2006 , PAGES =

    Miyake, Toshitsune , TITLE =. 2006 , PAGES =

    2006

  59. [68]

    1992 , PAGES =

    Lange, Herbert and Birkenhake, Christina , TITLE =. 1992 , PAGES =. doi:10.1007/978-3-662-02788-2 , URL =

    1992 doi

  60. [70]

    J.Ramanujan Math.Sco

    Schmidt,R , TITLE =. J.Ramanujan Math.Sco. , FJOURNAL =. 2002 , NUMBER =. doi:10.1007/s00209-021-02797-5 , URL =

    2002 doi

  61. [71]

    Hu, Yueke , TITLE =. Amer. J. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1353/ajm.2017.0004 , URL =

    2017 doi

  62. [72]

    , TITLE =

    Kudla, Stephen S. , TITLE =. An introduction to the. 2003 , MRCLASS =

    2003

  63. [73]

    , TITLE =

    Zagier, D. , TITLE =. Canad. Math. Bull. , FJOURNAL =. 1985 , NUMBER =. doi:10.4153/CMB-1985-044-8 , URL =

    1985 doi

  64. [74]

    and Stein, W

    Ribet, K. and Stein, W. , TITLE =. https://wstein.org/edu/2011/581g/ribet-stein/main.pdf , PAGES =. , MRCLASS =

    2011

  65. [75]

    Cremona, J. E. and Whitley, E. , TITLE =. Math. Comp. , FJOURNAL =. 1994 , NUMBER =. doi:10.2307/2153419 , URL =

    1994 doi

  66. [76]

    Periods of modular forms and imaginary quadratic base change , JOURNAL =

    Trifkovi\'. Periods of modular forms and imaginary quadratic base change , JOURNAL =. 2010 , NUMBER =. doi:10.4153/CMB-2010-047-0 , URL =

    2010 doi

  67. [77]

    2012 , PAGES =

    Getz, Jayce and Goresky, Mark , TITLE =. 2012 , PAGES =. doi:10.1007/978-3-0348-0351-9 , URL =

    2012 doi

  68. [78]

    Diem, Claus and Naumann, Niko , TITLE =. J. Ramanujan Math. Soc. , FJOURNAL =. 2003 , NUMBER =

    2003

  69. [79]

    Shimura, Goro , TITLE =. J. Math. Soc. Japan , FJOURNAL =. 1973 , PAGES =. doi:10.2969/jmsj/02530523 , URL =

    1973 doi

  70. [80]

    1997 , PAGES =

    Iwaniec, Henryk , TITLE =. 1997 , PAGES =. doi:10.1090/gsm/017 , URL =

    1997 doi

  71. [81]

    1982 , PAGES =

    Stevens, Glenn , TITLE =. 1982 , PAGES =

    1982

  72. [82]

    2005 , PAGES =

    Diamond, Fred and Shurman, Jerry , TITLE =. 2005 , PAGES =

    2005

  73. [83]

    Yin, Hongbo , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2022 , NUMBER =. doi:10.1090/tran/8556 , URL =

    2022 doi

  74. [84]

    2000 , PAGES =

    Lemmermeyer, Franz , TITLE =. 2000 , PAGES =. doi:10.1007/978-3-662-12893-0 , URL =

    2000 doi

  75. [85]

    Stevens, Glenn , TITLE =. Invent. Math. , FJOURNAL =. 1989 , NUMBER =. doi:10.1007/BF01388845 , URL =

    1989 doi

  76. [86]

    , Date-Added =

    Langlands, R.P. , Date-Added =. Unpublished note. https://publications.ias.edu/sites/default/files/a-ps.pdf , Title =

  77. [87]

    and Henniart, G

    Bushnell, C. and Henniart, G. , Date-Added =. The

  78. [88]

    Shu, Jie and Yin, Hongbo , TITLE =. Math. Ann. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s00208-022-02370-3 , URL =

    2023 doi

  79. [89]

    , TITLE =

    Elkies, Noam D. , TITLE =. Algorithmic number theory (. 1994 , MRCLASS =. doi:10.1007/3-540-58691-1\_49 , URL =

    1994 doi

  80. [90]

    , TITLE =

    Gonz\'alez, Josep and Lario, Joan-C. , TITLE =. Amer. J. Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1353/ajm.2001.0015 , URL =

    2001 doi

  81. [91]

    Monsky, Paul , TITLE =. Math. Z. , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF02570859 , URL =

    1990 doi

  82. [92]

    arXiv:2210.10730v2 , FJOURNAL =

    Alpöge, Levent and Bhargava, Manjul and Shnidman, Ari , TITLE =. arXiv:2210.10730v2 , FJOURNAL =. , NUMBER =. doi:, URL =

  83. [93]

    Majumdar, Dipramit and Shingavekar, Pratiksha , TITLE =. Proc. Indian Acad. Sci. Math. Sci. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s12044-023-00757-z , URL =

    2023 doi

  84. [94]

    , TITLE =

    Majumdar, Dipramit and Sury, B. , TITLE =. Int. J. Number Theory , FJOURNAL =. 2022 , NUMBER =. doi:10.1142/S1793042122500993 , URL =

    2022 doi

  85. [95]

    https://institucional.us.es/blogimus/en/2020/03/sylvesters-arithmetic-problem/ , FJOURNAL =

    Arias de Reyna, Juan , TITLE =. https://institucional.us.es/blogimus/en/2020/03/sylvesters-arithmetic-problem/ , FJOURNAL =. , NUMBER =. doi:, URL =

    2020

  86. [96]

    Cremona, J. E. , TITLE =. Math. Proc. Cambridge Philos. Soc. , FJOURNAL =. 1992 , NUMBER =. doi:10.1017/S0305004100075307 , URL =

    1992 doi

  87. [97]

    Li, Wen Ch'ing Winnie , TITLE =. Math. Ann. , FJOURNAL =. 1975 , PAGES =. doi:10.1007/BF01344466 , URL =

    1975 doi

  88. [98]

    and Tate, J

    Mazur, B. and Tate, J. and Teitelbaum, J. , TITLE =. Invent. Math. , FJOURNAL =. 1986 , NUMBER =. doi:10.1007/BF01388731 , URL =

    1986 doi

  89. [99]

    Cremona, J. E. , TITLE =. 1997 , PAGES =

    1997

  90. [100]

    Heninger, Nadia and Rains, E. M. and Sloane, N. J. A. , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 2006 , NUMBER =. doi:10.1016/j.jcta.2006.03.018 , URL =

    2006 doi

  91. [101]

    Calegari, Frank and Dimitrov, Vesselin and Tang, Yunqing , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1090/jams/1053 , URL =

    2025 doi

  92. [102]

    Stevens, Glenn , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 1985 , NUMBER =. doi:10.2307/2000098 , URL =

    1985 doi

  93. [103]

    , TITLE =

    Agashe, Amod and Ribet, Kenneth and Stein, William A. , TITLE =. Pure Appl. Math. Q. , FJOURNAL =. 2006 , NUMBER =. doi:10.4310/PAMQ.2006.v2.n2.a11 , URL =

    2006 doi

  94. [104]

    Burungale, Ashay and Flach, Matthias , TITLE =. Camb. J. Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/cjm.2024.v12.n2.a2 , URL =

    2024 doi

  95. [105]

    , Journal =

    Cesnavicius, K. , Journal =. The

  96. [106]

    , TITLE =

    Cesnavicius, K. , TITLE =. Compos. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1112/s0010437x18007273 , URL =

    2018 doi

  97. [107]

    Arithmetic algebraic geometry (

    Edixhoven, Bas , TITLE =. Arithmetic algebraic geometry (. 1991 , ISBN =. doi:10.1007/978-1-4612-0457-2\ _ 3 , URL =

    1991 doi

  98. [108]

    , Journal =

    Conrad, B. , Journal =

  99. [109]

    and Mazur, Barry , TITLE =

    Katz, Nicholas M. and Mazur, Barry , TITLE =. 1985 , PAGES =. doi:10.1515/9781400881710 , URL =

    1985 doi

  100. [110]

    , TITLE =

    Katz, Nicholas M. , TITLE =. Modular functions of one variable,. 1973 , MRCLASS =

    1973

  101. [111]

    and Rapoport, M

    Deligne, P. and Rapoport, M. , TITLE =. Modular functions of one variable,. 1973 , MRCLASS =

    1973

  102. [112]

    Hooley, Christopher , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1967 , PAGES =. doi:10.1515/crll.1967.225.209 , URL =

    1967 doi

  103. [113]

    Heath-Brown, D. R. , TITLE =. Quart. J. Math. Oxford Ser. (2) , FJOURNAL =. 1986 , NUMBER =. doi:10.1093/qmath/37.1.27 , URL =

    1986 doi

  104. [114]

    Ram , TITLE =

    Gupta, Rajiv and Murty, M. Ram , TITLE =. Invent. Math. , FJOURNAL =. 1984 , NUMBER =. doi:10.1007/BF01388719 , URL =

    1984 doi

  105. [115]

    Integers , FJOURNAL =

    Moree, Pieter , TITLE =. Integers , FJOURNAL =. 2012 , NUMBER =. doi:10.1515/integers-2012-0043 , URL =

    2012 doi

  106. [116]

    Lenstra, Jr., H. W. , TITLE =. Invent. Math. , FJOURNAL =. 1977 , PAGES =. doi:10.1007/BF01389788 , URL =

    1977 doi

  107. [117]

    Moree, Pieter , TITLE =. Funct. Approx. Comment. Math. , FJOURNAL =. 2008 , NUMBER =. doi:10.7169/facm/1229696559 , URL =

    2008 doi

  108. [118]

    Flath, Daniel , TITLE =. Math. Ann. , FJOURNAL =. 1979/80 , NUMBER =. doi:10.1007/BF01420163 , URL =

    1979 doi

  109. [119]

    Illinois J

    Wohlfahrt, Klaus , TITLE =. Illinois J. Math. , FJOURNAL =. 1964 , PAGES =

    1964

  110. [120]

    , Journal =

    Monsky, P. , Journal =

  111. [121]

    , Journal =

    Yin, H.B. , Journal =

Pith tools

Reviewed July 3, 2026 · model on record in the stance chip above.