A concise proof of the 8 case of Sylvester's conjecture

Hongbo Yin

REVIEW

A concise proof of the 8 case of Sylvester's conjecture

Hongbo Yin

Number Theory · Submitted 6 Oct 2026

Pith's verdict: A genuinely shorter route to the p ≡ 8 (mod 9) case, but its keystone torsion dichotomy is imported from the author's own unpublished preprint whose admitted errors sit in exactly the theorems used; the current version cannot be verified.

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arXiv ID2610.08015 (version 2)

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CitationHongbo Yin. "A concise proof of the 8 case of Sylvester's conjecture." arXiv:2610.08015 (2026). https://pith.science/paper/CQ3BEGPC

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Editor's note

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Colleague,

This paper is a new proof route, not a new theorem: Theorem 2.1 is the p ≡ 8 (mod 9) case already settled by Burungale and Tian, and the abstract says so. What is new is that the proof avoids Gross–Zagier, the √-3-division boundary, and Fermat isogeny, and instead derives the result from a 3-torsion ordinary trace plus a modular-function congruence. If the imported facts hold, that is a genuinely useful simplification.

The parts I could check independently look sound. The Galois-orbit argument in Proposition 3.1 via Milne's product formula is clean. The identification of E_ϖ[3] \ E_ϖ[√-3] with the six points (−ω^i t^2, ±ϖ√−3/2), and the cube-class contradiction in Proposition 4.4, are coherent. The formal-group lemma is standard, and the application with e = 3 < p−1 is legitimate for p ≥ 17. The p = 17, q = 13 example is explicit and the stated congruences check out. The theorem being already known is not a defect; the contribution is the route.

The soft spot is load-bearing, and the paper itself acknowledges it. Propositions 3.1 and 4.1 import [2, Theorems 4.2 and 8.6], including the claim that φ_c(τ_r) is √-3-torsion while φ(τ_r) has infinite order, plus the Shimura-reciprocity action that puts P in H_p L(ϖ). Footnote 3 says the condition on r in those theorems is reversed because the adelic subgroup U in [2] v3 was defined with the wrong component, and footnote 1 says the local-symbol calculation behind [2, Proposition 2.1] is wrong; both fixes are promised only in a future version. So the cited source does not currently certify the exact statements used. This is a verification gap with a plausible fix, not a demonstrated contradiction, but it blocks verification of the main theorem. The transfer of the Kronecker congruence from [15], originally in the 3p/3p^2 setting, to the level-9q parametrization is a second point where I would want the hypotheses stated explicitly; I do not see a failure, but the transfer is not shown.

Who this is for: specialists in cube sums and Heegner points who want to see whether Burungale–Tian's proof can be shortened. It is not for a general reader; without a corrected [2], the argument is an honest outline with stated gaps.

I would not send this manuscript out for formal refereeing in its current state. The right move is to return it and ask for the corrected [2], or for a version that proves the needed torsion dichotomy directly. Once that exists, the simplification deserves a serious referee.

Referee report

Significance

If correct, the paper would give a substantially simpler proof of the only open class of Sylvester's conjecture, avoiding the Gross-Zagier machinery and the √-3-division boundary used by Burungale-Tian. The internally verifiable pieces are coherent: the computation of E_ϖ[3]\E_ϖ[√-3] via the three-division polynomial, the cube-class analysis in Proposition 4.4, the formal-group lemma, and the p=17, q=13 example, including the factorization (4+3ω)(−2+5ω)=−23−ω and the concrete finite-field check [√-3][6]\tilde{B}≠O. The paper is also transparent about its main external dependency, declaring in footnotes that the relevant version of [2] has errors. The obstacle to acceptance is not internal inconsistency but verification: the keystone statements are imported from an unpublished preprint whose cited version is acknowledged to be defective in exactly the places on which the proof relies. With the corrected version of [2] available, the proof would likely be verifiable.

Major objections (2)

[§2.2 and §4.1, footnotes 1 and 3] The proof's keystone is imported from the author's unpublished preprint [2] v3, and that version is acknowledged in the manuscript not to certify the needed statements. Footnote 1 (sentence b36) states that the final local-symbol calculation behind [2, Proposition 2.1] is wrong, and footnote 3 (sentence b88) states that the condition on r in [2, Theorems 4.1, 4.2, 8.6] is reversed because the adelic subgroup U in [2] was defined with the wrong component. These imported facts — that φc(τr) is √-3-torsion while φ(τr) has infinite order, that φc(τr)=−φc(0)≠O, and the Shimura-reciprocity action used in (3.8) — are exactly what produce (4.1), (4.3), (4.4), and the field of definition in Proposition 3.1. As the text stands, Theorem 2.1 cannot be verified at its keystone. I am not claiming circularity: the dependence is openly declared. The fix is concrete: the author should provide the corrected statements of [2] in valid form (for example, by updating [2] and checking the parameters r, ϖ, q of (2.1)–(2.4) against them, or by stating the corrected theorems directly in this paper). The authors acknowledge this: “The same CM reciprocity calculation [2, proof of Theorem 4.1] gives [τr,1]σ3 = [Aτr,1]. 3The condition ofrin Theorems 4.1, 4.2 and 8.6 of the version 3 of [2] is reversed due to the wrong condition of the adelic congruence groupU(we should defineUusing matrix componentainstead ofd).”
[§4.1, Proposition 4.2, equations (4.5)–(4.6)] The reduction congruence ~S_{p,r}=~S_r^{Fr_p} is imported from [15, proof of Proposition 2.4] and asserted for the level-9q parametrizations of this paper. The cited source works with the forms 3p and 3p^2, while the present application concerns level-9q paramatrizations, and the manuscript gives no derivation or precise statement of how the congruence transfers to this setting. This congruence is load-bearing: it is the bridge that identifies the reduction of the conjugates ~P^σ with a fixed Frobenius power of ~P', and through (4.15)–(4.16) it underlies the non-√-3-divisibility in Proposition 4.4. The author should either point to the exact statement in [15] that covers this level and parametrization, or prove the needed congruence here.

Minor objections (3)

[§2.1, equation (2.1)] The existence of infinitely many q satisfying (2.1) is cited from [4, Lemma 2.1], and [4] is currently an arXiv preprint. Since the statement is a standard Chebotarev consequence, a short proof or a stable published reference would make the paper less dependent on yet another preprint.
[§6] The example would be easier to check if the origin of the polynomials A(T) and B(T) in (6.3)–(6.4) were explained, since these expressions are presented without derivation. The numerical value is useful as a check, but it is not part of the proof.
[Throughout] There are small typographical issues: footnote 1 contains "a error", and Section 6 ends a sentence with the stray comma in "Here [H17 : K] = 6,.". These do not affect the mathematics.

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The paper

The paper claims

For every prime $p \equiv 8 \pmod{9}$, the paper shows two twisted traces of a CM point on a modular curve have infinite order, giving a concise proof of the 8-case of Sylvester's conjecture, namely that $p$ and $p^2$ are sums of two…

The reading

This paper claims to close the last open case of Sylvester's 150-year-old conjecture: every prime $p \equiv 8 \pmod{9}$ is a sum of two nonzero rational cubes, and so is $p^2$. Since $p$ is a cube sum exactly when the elliptic curve $E_p : y^2 = x^3 + \frac{p^2}{4}$ has positive rank over $\mathbb{Q}$, the paper constructs a point with complex multiplication (a CM point) on a modular curve of level $9q$, forms its ordinary trace into the cubic field $L(\varpi,p)$, and proves that this trace is $3$-torsion but not $\sqrt{-3}$-divisible. That failure of division forces two twisted components $z_1, z_2$ to be nonzero, and their reduction to the origin above $p$ makes them points of infinite order by the formal-group torsion bound. The author offers this as a much shorter route to the case-8 result than the previous proof of Burungale–Tian, and gives the explicit example $p=17$, $q=13$; if correct, it completes Sylvester's conjecture together with the author's earlier 4,7-case proof.

What carries the argument

The load-bearing object is the pair of twisted Galois traces of a CM point. The point $P = \varphi_c(p\tau_r)$ lies on $E_{\varpi}$, the conjugate modular parametrization of level $N = 9q$, where $\tau_r = -\frac{1}{3}(\omega+r)$ with $r^2 - r + 1 \equiv 0 \pmod{3q}$ and $-r \equiv \omega \pmod{\varpi}$; from $P$ the paper forms the total trace $z_0$ of (2.6) and the eigen-components $z_1, z_2$ of (2.7)–(2.8) on which $\sigma_p$ acts by $\omega$ and $\omega^2$. Proposition 3.1 identifies the Hecke orbit of $\tau_r$ with its Galois orbit over $H_{3p}L(\varpi)$ and uses $a_p(f^c) = 0$ to show $z_0$ is $3$-torsion; Proposition 4.2 pairs the reductions under the Kronecker congruence $\widetilde{p\tau_r} = \widetilde{\tau_r}^{\, \mathrm{Fr}_p}$, so that $\widetilde{z_0} = [\omega^i][m] \widetilde{P'}^{\, \mathrm{Fr}_p}$, and Lemma 4.3 proves $[\sqrt{-3}m] \widetilde{P'} \neq O$ from the cube-class identity $y(B) + q/2 \equiv \varpi^{-\epsilon} \pmod{K^{\times 3}}$ and the group structure $E_q(k) \simeq (\mathbb{Z}/(p+1)\mathbb{Z})^2$. These pieces show $z_0$ is $3$-torsion outside the $\sqrt{-3}$-kernel, which a cube-class obstruction makes incompatible with $\sqrt{-3}$-division in $L(\varpi)$; Proposition 5.1 and Lemma 5.2 then convert that into the infinite order of $z_1, z_2$.

Extended reading notes

Core claim

Theorem 2.1 asserts that for every prime $p \equiv 8 \pmod{9}$, both points $z_1$ and $z_2$ — and hence $R_1 = \varphi_1(z_1)$ and $R_2 = \varphi_2(z_2)$ — have infinite order, so that $p$ and $p^2$ are each sums of two nonzero rational cubes. The proof's discovery is that a single trace dichotomy carries the entire rank conclusion: the ordinary trace $z_0 = \mathrm{Tr}_{H_p L(\varpi)/L(\varpi,p)}(P)$ of the CM point $P = \varphi_c(p\tau_r)$ is $3$-torsion by the Galois-orbit and Hecke-orbit calculation (Proposition 3.1), yet lies outside the kernel of $\sqrt{-3}$ by the Kronecker-congruence and finite-field divisibility argument (Proposition 4.4). The identities (5.1)–(5.2) then force $z_1$ and $z_2$ to be nonzero (Proposition 5.1), and the formal-group lemma (Lemma 5.2), with absolute ramification index $e = 3 < p-1$, rules out torsion in their reduction to the origin. The chain yields infinite order for every prime in the class, not only for the worked example $p = 17$.

Load-bearing premise

The load-bearing premise is that the results the proof imports from the author's own unpublished preprint [2] (version 3) hold for the chosen $r$, $\varpi$, $q$: that $\varphi_c(\tau_r)$ is $\sqrt{-3}$-torsion while $\varphi(\tau_r)$ has infinite order, together with the Shimura-reciprocity actions (3.8), the transformation law (3.1), and the cubic-function identity (4.4). The paper's footnotes (Sections 2.1, 3, 4.1) admit that the cited version of [2] has a reversed hypothesis on $r$, a misdefined adelic subgroup $U$, and a wrong local-symbol calculation, with corrections promised only in a future version; if the corrected statements do not hold for these parameters, Propositions 3.1, 4.1, 4.4 and Theorem 2.1 collapse. A secondary fragile transfer is the Kronecker congruence of [15], applied to the level-$9q$ parametrizations in the $3p$ and $3p^2$ setting.

Editorial extensions

If this is right

Every prime $p \equiv 8 \pmod{9}$, and also $p^2$, is a sum of two nonzero rational cubes; together with the 4,7 cases this completes Sylvester's conjecture.
Each curve $E_p : y^2 = x^3 + \frac{p^2}{4}$ for $p \equiv 8 \pmod{9}$ has positive rank over $\mathbb{Q}$, matching the parity predictions of Birch and Stephens.
The case-8 conclusion is reached without the Gross–Zagier formula, the $\sqrt{-3}$-division boundary, or the Fermat isogeny; an ordinary trace, a modular-function congruence, and a formal-group bound suffice.
For $p = 17$ the paper supplies explicit rational coordinates for $z_0, z_1, z_2$ (Section 6), so the non-torsion claim is directly checkable at the smallest prime of the class.
The nonvanishing of $z_1, z_2$ is obtained with no height computation, so the same trace dichotomy is claimed uniformly across the entire prime class.

Reading between the lines

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

Because the author's own footnotes (Sections 2.1, 3, and 4.1) state that the cited version of the companion preprint has a reversed hypothesis on $r$, a misdefined adelic subgroup, and a wrong local-symbol calculation, the theorem as printed is conditional on the promised corrections to that preprint; this is my reading of the paper's caveats, not a judgement that the theorem is false.
The same trace dichotomy might apply to $p \equiv 1 \pmod{9}$, where Selmer's bound allows rank up to 2 but the root number is $+1$; a modified ordinary-trace argument could in principle produce positive rank there, though the paper does not claim this.
The auxiliary prime $q$ enters only through the non-cube condition (2.1), which suggests a $p$-adic variant that reads the $\sqrt{-3}$-divisibility obstruction directly from the formal logarithm of $E_{\varpi}$ over $\mathbb{Q}_p$, giving a purely local proof of the nonvanishing; the paper does not pursue this direction.
If the construction is uniform in $p$, the same CM-point trace should yield explicit rational cube-sum identities for composite $n$ whose prime factors lie in the classes $4, 7, 8 \pmod{9}$ with root number $-1$, a testable extension beyond primes that the paper leaves untouched.

Original abstract

The 150 years old Sylvester's conjecture asserts that every prime $p\equiv 4,7,8\mod 9$ is the sum of two rational cubes. Recently, a proof of this old problem is obtained by the work of mine for the case 4,7 \cite{Yin} and the work of Burungale-Tian for the case 8 \cite{BT}. In this paper we give a shorter proof by deep mining my result in \cite{Yin} combining with the ideas of my previous work on case 8 in \cite{Yin8}. In particular, we use the ordinary trace and the Kronecker congruence of modular functions, but do not use the $\sqrt{-3}$-division boundary and Fermat isogeny as in \cite{BT}.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-07: 1 request, 5,107 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.98 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The 150 years old Sylvester's conjecture asserts that every prime $p\equiv 4,7,8\mod 9$ is the sum of two rational cubes. Recently, a proof of this old problem is obtained by the work of mine for the case 4,7 \cite{Yin} and the work of Burungale-Tian for the case 8 \cite{BT}. Burungale-Tian's paper essentially uses the results of paper \cite{Yin}, but does not make full use of the power. Their proof for the case $8$ is complicated and hard to follow. In this paper we give a much simpler and direct proof by deep mining my result in \cite{Yin} combining with the ideas of my previous work on case 8 in \cite{Yin8}. In particular, we use the ordinary trace and the Kronecker congruence of modular functions, but do not use the Gross-Zagier formula, $\sqrt{-3}$-division boundary and Fermat isogeny as in \cite{BT}.This concise proof provides a perfect end to this old concise conjecture.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The 150 years old Sylvester's conjecture asserts that every prime $p\equiv 4,7,8\mod 9$ is the sum of two rational cubes.	background 1.00	0.03	0.06	asserted 0.83
s1	Recently, a proof of this old problem is obtained by the work of mine for the case 4,7 \cite{Yin} and the work of Burungale-Tian for the case 8 \cite{BT}.	background 0.96	0.72	0.07	asserted 1.00
s2	Burungale-Tian's paper essentially uses the results of paper \cite{Yin}, but does not make full use of the power.	gap 0.60	0.17	0.04	asserted 1.00
s3	Their proof for the case $8$ is complicated and hard to follow.	gap 0.54	0.11	0.03	asserted 1.00
s4	In this paper we give a much simpler and direct proof by deep mining my result in \cite{Yin} combining with the ideas of my previous work on case 8 in \cite{Yin8}.	contribution 0.96	0.91	0.48	asserted 1.00
s5	In particular, we use the ordinary trace and the Kronecker congruence of modular functions, but do not use the Gross-Zagier formula, $\sqrt{-3}$-division boundary and Fermat isogeny as in \cite{BT}.This concise proof provides a perfect end to this old concise conjecture.	method 0.84	0.88	0.51	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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figures and tables
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levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Discussion

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