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Every prime congruent to 4 or 7 modulo 9 is the sum of two rational cubes.

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Proves that every prime p ≡ 4,7 mod 9 is a sum of two rational cubes, resolving 2/3 of Sylvester's conjecture via BSD progress and the solved Unbounded Denominators Conjecture.

T0 review reviewed 2026-06-29 challenge →

load-bearing objection Yin claims to prove the 4 and 7 mod 9 cases of Sylvester's conjecture by applying recent BSD and unbounded denominators results, but the full details remain unchecked. the 1 major comments →

arxiv 2605.25917 v3 pith:FDYKCUFP submitted 2026-05-25 math.NT

A proof of the $4,7$ cases of Sylvester's conjecture on cube sums

Hongbo Yin This is my paper
classification math.NT
keywords Sylvester conjecturesum of two rational cubeselliptic curvesManin-Stevens constantsUnbounded Denominators Conjectureprimes modulo 9Birch and Swinnerton-Dyer conjecture
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 paper proves that every prime p congruent to 4 or 7 modulo 9 can be expressed as the sum of two rational cubes. This resolves two-thirds of Sylvester's conjecture from 1879. The argument shows that Manin-Stevens constants attached to families of elliptic curves are units, which implies the existence of the required rational points. Recent results on the full Birch and Swinnerton-Dyer conjecture for rank-zero curves and the solution of the Unbounded Denominators Conjecture supply the needed unit property and the invariance of certain cubic roots of modular functions. A sympathetic reader would care because the result converts an old question about integer equations into a statement about arithmetic properties of elliptic curves.

Core claim

The central claim is that for every prime p congruent to 4 or 7 modulo 9 the equation a^3 + b^3 = p admits solutions a, b in the rational numbers. The proof proceeds by associating to each such p a family of elliptic curves whose Manin-Stevens constants are shown to be units, using the full BSD conjecture for rank-zero curves together with the Unbounded Denominators Conjecture to guarantee that the cubic roots of certain modular functions remain invariant under appropriate congruence subgroups.

What carries the argument

Manin-Stevens constants of families of elliptic curves attached to the equation x^3 + y^3 = p z^3, shown to be units so that the curves possess rational points of infinite order.

Load-bearing premise

The Manin-Stevens constants of the relevant families of elliptic curves are units.

What would settle it

A single prime p congruent to 4 or 7 modulo 9 together with a proof that the equation a^3 + b^3 = p has no rational solutions a, b would falsify the claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Every prime congruent to 4 or 7 modulo 9 admits a representation as a sum of two rational cubes.
  • Two-thirds of the cases in Sylvester's conjecture on cube sums are settled.
  • The method produces an unconditional proof for these congruence classes and a conditional proof under GRH or Artin's primitive-root conjecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same circle of ideas could be tested on composite integers whose prime factors all lie in the settled residue classes.
  • If the remaining residue classes modulo 9 can be treated by analogous techniques, the full Sylvester conjecture would follow.
  • Explicit bounds on the height of the rational solutions might be extracted from the size of the Manin-Stevens constants once they are known to be units.
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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 that every prime p congruent to 4 or 7 modulo 9 is the sum of two rational cubes, resolving two-thirds of Sylvester's conjecture. The argument applies recent results on the full BSD conjecture for rank-zero elliptic curves [BF] to conclude that the Manin-Stevens constants of certain families are units, and invokes the solution of the Unbounded Denominators Conjecture [CDT] to establish that cubic roots of modular functions are invariant under specified congruence subgroups; conditional proofs assuming GRH for number fields or Artin's primitive root conjecture are also supplied.

Significance. If the applications of [BF] and [CDT] hold, the result would be a major contribution to arithmetic geometry by settling a 150-year-old Diophantine problem for a substantial infinite family of primes, linking elliptic-curve arithmetic, modular forms, and rational points.

major comments (1)
  1. [Abstract and §1] The central claim rests on the deduction that Manin-Stevens constants are units, which is asserted to follow from [BF]; without an explicit, self-contained verification of this step inside the manuscript (including the precise families of curves and the relevant L-values or Sha computations), the load-bearing step cannot be assessed for correctness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater explicitness in the application of [BF]. We address the major comment below and will incorporate revisions to make the deduction fully verifiable.

read point-by-point responses

  1. Referee: [Abstract and §1] The central claim rests on the deduction that Manin-Stevens constants are units, which is asserted to follow from [BF]; without an explicit, self-contained verification of this step inside the manuscript (including the precise families of curves and the relevant L-values or Sha computations), the load-bearing step cannot be assessed for correctness.

    Authors: We agree that the manuscript currently asserts the deduction from [BF] without a fully self-contained verification. In the revised version we will add a dedicated subsection (in §2) that: (i) identifies the precise families of elliptic curves arising from the primes p ≡ 4,7 mod 9; (ii) recalls the relevant statements of [BF] on the full BSD conjecture for rank-zero curves; (iii) records the explicit L-values at s=1 and the vanishing of Sha for these curves; and (iv) confirms that the resulting Manin-Stevens constants are units. This addition will not alter the logical structure of the proof but will allow direct assessment of the step. We also note that the conditional proofs (via GRH or Artin’s conjecture) remain unchanged and do not rely on this step. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's main argument deduces the result from external theorems: Full BSD for rank-0 curves via [BF] to obtain unit Manin-Stevens constants, and the solution of the Unbounded Denominators Conjecture via [CDT] to establish invariance of cubic roots of modular functions. These are independent results from other authors, not self-citations or internal definitions. The derivation chain therefore rests on externally verified benchmarks rather than reducing any prediction or uniqueness claim to quantities defined inside the present work. A conditional proof under GRH or Artin's conjecture is also offered, again relying on external assumptions. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of two external results: progress toward the full BSD conjecture for rank-0 curves and the solution of the Unbounded Denominators Conjecture. No free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Recent progress on the Full BSD conjecture for rank-0 elliptic curves implies that Manin-Stevens constants of certain families are units
    Invoked to deduce unit constants for the elliptic curves arising in the proof.
  • domain assumption The Unbounded Denominators Conjecture has been solved, allowing cubic roots of modular functions to be shown invariant under certain congruence subgroups
    Used to establish the required invariance properties.

reviewed 2026-06-29 · how reviews work

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Cite this review

Pith. "Pith review of A proof of the $4,7$ cases of Sylvester's conjecture on cube sums." pith.science (2026). https://pith.science/paper/FDYKCUFP

@misc{pith2026260525917,
  author       = {Pith},
  title        = {Pith review of: A proof of the $4,7$ cases of Sylvester's conjecture on cube sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDYKCUFP}},
  note         = {Machine review of arXiv:2605.25917}
}
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abstract

In this paper, we prove that every prime $p$ which is congruent to $4,7$ modulo $9$ is the sum of two rational cubes. This is $2/3$ of Sylvester's conjecture which has a history of nearly 150 years since 1879. In the proof, we use recent progress on Full BSD conjecture of rank $0$ elliptic curves in \cite{BF} to deduce that the Manin-Stevens constants of some families of elliptic curves are units. We also use recent solutions of Unbounded Denominators Conjecture in \cite{CDT} to prove that some cubic roots of modular functions are invariant under some congruence subgroups. Instead of using the Unbounded Denominators Conjecuture, we also give another conditional proof assuming the GRH for number fields or Artin's primitive root conjecture for arithmetic progressions.

discussion (0)

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Reference graph

Works this paper leans on

11 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    arXiv:2210.10730v2 , FJOURNAL =

    [ABS] Levent Alp¨ oge, Manjul Bhargava, and Ari Shnidman. Integers expressible as the sum of two rational cubes.arXiv:2210.10730v2, with an appendix by Ashay Burungale and Christopher Skinner. [AdR] Juan Arias de Reyna. Sylvester’s arithmetic problem. https://institucional.us.es/blogimus/en/2020/03/sylvesters-arithmetic- problem/. [AL78] A. O. L. Atkin an...

    work page arXiv 2020

  2. [2]

    The Manin-Stevens constant in the semistable case

    [Ces] K. Cesnavicius. The Manin-Stevens constant in the semistable case. arXiv:1604.02165v1. [Ces18] K. Cesnavicius. The Manin constant in the semistable case.Compos. Math., 154(9):1889–1920,

    work page internal anchor Pith review Pith/arXiv arXiv 1920

  3. [3]

    Deligne and M

    [DR73] P. Deligne and M. Rapoport. Les sch´ emas de modules de courbes elliptiques. InModular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 349, pages 143–316. Springer, Berlin-New York,

    1972

  4. [4]

    Dasgupta and J

    [DV09] S. Dasgupta and J. Voight. Heegner points and sylvester’s conjecture.Arithmetic Geometry: Clay Mathematics Institute Summer School, Arithmetic Geometry, July 17-August 11, 2006, Mathematisches Institut, Georg-August-Universit¨ at, G¨ ottingen, Germany, 8:91,

    2006

  5. [5]

    On the Manin constants of modular elliptic curves

    36 HONGBO YIN [Edi91] Bas Edixhoven. On the Manin constants of modular elliptic curves. InArith- metic algebraic geometry (Texel, 1989), volume 89 ofProgr. Math., pages 25–39. Birkh¨ auser Boston, Boston, MA,

    1989

  6. [6]

    [Elk94] Noam D. Elkies. Heegner point computations. InAlgorithmic number theory (Ithaca, NY, 1994), volume 877 ofLecture Notes in Comput. Sci., pages 122–

    1994

  7. [7]

    Katz.p-adic properties of modular schemes and modular forms

    [Kat73] Nicholas M. Katz.p-adic properties of modular schemes and modular forms. In Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 350, pages 69–190. Springer, Berlin-New York,

    1972

  8. [8]

    Translated from the 1976 Japan- ese original by Yoshitaka Maeda. [Mon] P. Monsky. The sum of two cubes problem—an approach that’s classroom friendly.arXiv:2309.00162v1. [Mor08] Pieter Moree. On primes in arithmetic progression having a prescribed primitive root. II.Funct. Approx. Comment. Math., 39(part 1):133–144,

    work page arXiv 1976

  9. [9]

    [MS22] Dipramit Majumdar and B. Sury. Cyclic cubic extensions ofQ.Int. J. Number Theory, 18(9):1929–1955,

    1929

  10. [10]

    Reprint of the 1971 original, Kanˆ o Memorial Lectures,

    1971

  11. [11]

    [Sil94] J

    Corrected reprint of the 1986 original. [Sil94] J. H. Silverman.Advanced topics in the arithmetic of elliptic curves, volume 151 ofGraduate Texts in Mathematics. Springer-Verlag, New York,

    1986

This paper was first reviewed by grok-4.3 on June 29, 2026.