A proof of Sylvester's conjecture

Ashay Burungale; Ye Tian

REVIEW

A proof of Sylvester's conjecture

Ashay Burungale, Ye Tian

Number Theory · Submitted 14 Sep 2026

Pith's verdict: A serious and important paper that likely resolves Sylvester's conjecture, but the proof's dependence on the authors' own very recent converse theorem [9] is a load-bearing vulnerability.

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arXiv ID2609.14893 (version 2)

Pith ID23NXQQCT, Pith's permanent record of this paper and its review

CitationAshay Burungale, Ye Tian. "A proof of Sylvester's conjecture." arXiv:2609.14893 (2026). https://pith.science/paper/23NXQQCT

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Editor's note

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This is the real thing. Burungale and Tian prove the remaining p ≡ 8 mod 9 case of Sylvester's conjecture, completing a problem from 1879. The central new idea—the division boundary technique to recover arithmetic information from a vanishing Hecke trace—is genuine and clever. The Frobenius calculation using the cubic residue symbol, the passage from the boundary to cubic character components via integral group-ring identities, and the identification with the auxiliary Heegner point are all well executed. The paper is long but clearly organized; the examples in the appendix help ground the geometry.

The soft spot is exactly what the stress test flags: the argument depends critically on the rank zero 3-converse theorem [9] to get L(1,E_{D_{q,j}}) ≠ 0. That theorem is the authors' own 2026 preprint, and its proof may itself rely on as-yet-unverified Iwasawa-theoretic results. If [9] has an undetected gap, the factorization in Proposition 2.3 collapses and the whole proof breaks. The paper also uses Yin's recent preprint for the 4 and 7 cases, though that is less central. The cubic residue condition (1.3) and Chan's descent matrix are checked generically and seem solid. The division boundary argument itself appears internally consistent and does not reuse the converse theorem.

So the paper is genuinely new and important if the external inputs hold up. It deserves a serious referee who can assess [9] as well. The authors are honest about their dependencies, and the proof architecture is transparent.

For whom: number theorists working on BSD, Heegner points, and cube sums. The paper is technically demanding but the rewards are high.

My call: this should go to peer review. The editor should assign a referee who can evaluate both the current paper and the converse theorem it leans on. The paper is not ready for acceptance until [9] is verified or the proof is made independent, but it is far too substantial to desk-reject.

What would settle it

Go and compute: for an explicit admissible pair ($p \equiv 8 \pmod 9$, $q \equiv 4 \pmod 9$, $(p/\varpi)_3 \neq 1$) beyond the worked examples, reduce the explicit Fermat division point $\bar{R}_q$ at the prime above $p$ and check Proposition 6.7's prediction $[p+1]\bar{R}_q = \varepsilon_{q,p} T$ on $E_{\bar\varpi}$, equivalently that the division boundary equals $\pm(0, \varpi/2)$; the appendix's pairs ($q = 13$, $q = 31$ with $p = 17$, and $p = 107$) are the seed cases, and a single pair where the reduced boundary vanishes would falsify the load-bearing step. Independently, numerically evaluate $L(s, E_p)$ near $s = 1$ for the smallest primes of the class (17, 107, 269, and so on): the theorem predicts a simple zero, and a computed order other than one for any of them would refute Theorem 1.1.

Referee report

Major objections (2)

[§3.2, Proposition 3.2] The non‑vanishing of L(1, E_{D_{q,j}}) depends critically on the rank‑zero 3‑converse theorem of Burungale–Tian [9]. That theorem is very recent (2026) and its full proof may rely on other unverified Iwasawa‑theoretic inputs. Since the entire argument collapses if [9] has a gap, the authors must discuss the current status of [9] (e.g., has it been accepted for publication? Is there an independent verification?). If the journal policy permits reliance on arXiv preprints, a clear statement of the assumption should be added.
[§3.1, Lemma 3.1] The matrix R^T_j is taken from Chan [11]. While the paper reproduces the explicit entries, the theorem of Chan (Theorem 1.1 of [11]) is itself a result from 2024. The authors should confirm that the notation and normalisation in [11] match those used here (e.g., the sign convention for the cubic residue symbol, the definition of δ_m). A brief remark or a reference to the exact equation in [11] would strengthen this step.

Minor objections (10)

[§1.6.7, Frobenius conventions] The table of four Frobenius operations is very helpful, but the entry for 'Fr^(λ)_p' could be made more explicit: the action [¯λ/λ] is defined as the endomorphism of the λ‑kernel; it would be clearer to state that u_λ = ¯λ/λ equals −ω².
[§6.1, Lemma 6.1] In the displayed formula for ȷ^{-1}_a, the expression contains a factor (1+2ω) which equals ωλ; this is correct, but a brief remark that ωλ = 1+2ω would help readability.
[§7.2, Proposition 7.3] Equation (7.2) asserts that all φ^c‑images of the conductor‑p orbit have the same reduction, equal to Ȳ_{Ē_ϖ}/F_{p²}(Q̄_q). The proof uses the q‑expansion principle and the defining property of φ^c. A small clarification of why φ̅^{(p)} = φ̅^c would be welcome (e.g., because the non‑trivial automorphism of K/Q induces the p‑power map on the residue field).
[§6.3, Proposition 6.7] The sign ε_{q,p} is given by a table. It might be useful to state the formula as ε_{q,p} = −δ(η_q, α_q(p)), where δ(·,·) is defined in (6.20). Currently the table is clear but a short formula would unify the cases.
[§8.2, Lemma 8.4] The statement that [ϖω] ≠ 1 in F_q^× / F_q^{×3} is justified by the fact that K(ζ_9) would otherwise be contained in F_q, contradicting ramification. This is correct, but the argument could be cited more directly: because F_q/K is ramified at q while K(ζ_9)/K is unramified there.
[§9.3, Proposition 9.8] The action of δ₃ on φ^c‑images is given as [ω²] after the p‑modification. The proof refers to Lemma 7.5, but that lemma only treats ∆_p. The computation of χ(δ₃) should be expanded: the principal idele ω and the local class at 3 produce a factor ω; the product gives the stated identity. A brief justification would remove a small obscurity.
[§10.1, Lemma 10.2] The argument that the incoherent quaternion algebra is split at every finite place uses the non‑vanishing of the Heegner point. This is a standard implication, but the reader would benefit from a reference to the Tunnell–Saito criterion as presented in [10, §3A]. The sentence “The nearby coherent quaternion algebra … is therefore B ≃ M₂(Q)” could be expanded to explain why switching the archimedean invariant yields global splitness.
[§7.4, Definition 7.8] The notation ∂^c_{q,p,P} uses P for a prime above p; later in Definition 7.11 it is dropped because independence is proved. It would be clearer to state the independence immediately after the definition, rather than waiting for Proposition 7.10.
[§A.2] In the example q=13, the point Q₁₃ is given with co‑ordinates (−2ω−7)t^{−2} and (9ω+7)/2. The check that ℓ_ϖ̄(R₁₃)=Q₁₃ could be indicated (it follows from the cubic identity). The table of ε_{q,p} for p=17 and p=107 is useful.
[§A.3] The example q=31 shows a non‑trivial cube factor v³, reinforcing that the Kummer class rather than the explicit cube root matters. This is a good illustration.

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The paper

The paper claims

This paper proves the final case of Sylvester's 1879 cube-sum conjecture: every prime p ≡ 8 (mod 9) is a sum of two rational cubes, by showing the associated elliptic curves have analytic rank one.

The reading

The paper completes Sylvester's conjecture, the 1879 claim that every prime $p \equiv 4, 7, 8 \pmod 9$ is a sum of two rational cubes. The classes 4 and 7 were announced in 1994 and recently proved in full; this paper settles the remaining class $p \equiv 8 \pmod 9$, where the quadratic field $\mathbb{Q}(\omega)$ is inert and the natural Hecke-orbit trace vanishes. The proof establishes $\operatorname{ord}_{s=1} L(s, E_m) = 1$ for $m = p$ and $m = p^2$ on $E_m : y^2 = x^3 + m^2/4$; the Gross-Zagier and Kolyvagin theorems then force Mordell-Weil rank one, and classical descent exhibits $p$ and $p^2$ as sums of two rational cubes. The significance is threefold: a 150-year-old Diophantine question is closed, the Birch-Swinnerton-Dyer prediction of a simple zero is confirmed for an infinite family, and a new device — the division boundary, the trace of the first $\lambda$-divisions of the CM orbit with $\lambda = 1 - \omega$ — recovers arithmetic information precisely where the unweighted Hecke trace vanishes. The core content is that this boundary is provably non-zero, detected at $p$ by Frobenius.

What carries the argument

The carrying object is the division boundary (Definition 7.11). For the conductor-$p$ Hecke orbit $\{x_u : u \in C_p\}$ of a CM point on the degree-three modular cover $X_\Gamma$ of $X_0(9q)$, with projected images $Q_u = \varphi^c(x_u)$ on $E_\varpi$, the paper first chooses points $R_u$ with $[\lambda] R_u = Q_u$ ($\lambda = 1 - \omega$, $\omega$ a primitive cube root of unity), and only then forms the trace $\partial^c_{q,p} = \sum_{u \in C_p} R_u$. Because the unweighted trace $\sum_u Q_u$ vanishes (the Hecke eigenvalue $a_p(g_q^c)$ is zero), this division-then-trace order is what retains information; the boundary lands in the $\lambda$-torsion $E_\varpi[\lambda](K)$, is independent of all choices, and is computed in the special fibre: comparing the $p^2$-power Frobenius on the Fermat co-ordinates with the CM endomorphism $[-p\alpha_q(p)]$ gives $\partial^c_{q,p} = \varepsilon_{q,p} T^\sharp_E \neq O$, non-vanishing exactly because the cubic residue symbol $\alpha_q(p) = (p/\varpi)_3$ is non-trivial. The companion algebraic identity is the integral group-ring relation $\lambda D = N_G - \Pi$ with $D = (N_G - \Pi)/\lambda$ integral, which forces the norm $S = N_G Z$, a point of $E_\varpi[3] \setminus E_\varpi[\lambda]$, to be $\lambda$-divisible over $F_q$ if any cubic component $\Pi Z$ vanished, a contradiction to the explicit Kummer class $[\varpi\omega] \neq 1$ in $F_q^\times / F_q^{\times 3}$. In short, the division boundary replaces the indivisibility of a distinguished Heegner point — the tool of earlier approaches — with a torsion-valued invariant of an orbit whose trace is zero.

Extended reading notes

Core claim

The paper's central claim (Theorem 1.1) is that for every prime $p \equiv 8 \pmod 9$, $\operatorname{ord}_{s=1} L(s, E_p) = \operatorname{ord}_{s=1} L(s, E_{p^2}) = 1$ for the curves $E_m : y^2 = x^3 + m^2/4$; together with the completed proof for the classes 4 and 7, this yields Sylvester's conjecture (Theorem 1.2). To get there the authors choose an auxiliary prime $q \equiv 4 \pmod 9$ whose cubic residue symbol $(p/\varpi)_3$ is non-trivial (such primes exist in density $1/9$ by Chebotarev), and factor a self-dual Rankin-Selberg convolution as $L(s, g_q \times \chi_{q,1}) = L(s, E_p) L(s, E_{qp^2})$ and $L(s, g_q \times \chi_{q,2}) = L(s, E_{p^2}) L(s, E_{qp})$. An explicit 3-isogeny descent computation together with the paper's own rank-zero 3-converse theorem for CM elliptic curves shows the complementary central values $L(1, E_{qp})$, $L(1, E_{qp^2})$ are non-zero, so the YZZ generalization of the Gross-Zagier formula reduces the theorem to proving that a cubic component of an auxiliary Heegner point is non-torsion. At that point the central new step appears: the unweighted Hecke trace of the conductor-$p$ CM orbit vanishes, so the authors take a first $\lambda$-division before tracing, $\lambda = 1 - \omega$; the resulting division boundary $\partial^c_{q,p}$ is a $\lambda$-torsion point shown to be non-zero by comparing the $p^2$-power Frobenius on Fermat co-ordinates with the CM endomorphism $[-p \alpha_q(p)]$, where the non-trivial residue symbol is exactly what is detected. Integral group-ring identities $\lambda D = N_G - \Pi$ transfer this non-zero torsion to the cubic component, and the Gross-Zagier formula turns it into $L'(1, g_q \times \chi_{q,j}) \neq 0$, completing the analytic-rank-one proof.

Load-bearing premise

The load-bearing premise is a very recent 'rank-zero 3-converse' theorem for complex-multiplication elliptic curves: it asserts that the auxiliary curves $E_{qp}$ and $E_{qp^2}$ have non-zero central $L$-value $L(1, E_{qp^j})$ once their 3-Selmer groups are trivial, and the whole proof for $p \equiv 8 \pmod 9$ collapses if that theorem, or the recently completed proof of the 4 and 7 classes, is unsound.

Editorial extensions

If this is right

Sylvester's conjecture is now a theorem: every prime $p \equiv 4, 7, 8 \pmod 9$ is a sum of two rational cubes, with the new case $p \equiv 8 \pmod 9$ producing such representations for both $p$ and $p^2$.
For every prime $p \equiv 8 \pmod 9$, the elliptic curves $E_p$ and $E_{p^2}$ have Mordell-Weil rank exactly one and a finite Tate-Shafarevich group, so the analytic-to-algebraic rank agreement predicted by BSD holds for this infinite family.
The division boundary supplies a replacement for Heegner-point indivisibility in situations where a Hecke orbit's unweighted trace vanishes; the paper states explicitly that it expects this mechanism to extend beyond these curves and to admit higher analogues in conductor towers.
For any $p \equiv 8 \pmod 9$, the proof constructs the auxiliary data ($q$, $\varpi$, the Heegner point) through Chebotarev, so the cube representations are in principle computable; the appendix makes this explicit for $p = 17$ with $q = 13$ and $q = 31$.
pith_inferences: This string is not an inference and must be ignored.

Reading between the lines

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

The division-then-trace device should generalize to other isogenies: replacing $\lambda$ by a generator of an $\ell$-power isogeny kernel would attach a higher 'division boundary' to any Hecke orbit with vanishing trace, turning the paper's conjectured conductor-tower analogue into a tower of torsion-valued invariants rather than a single point.
Read cohomologically, the boundary is the image of the norm-zero class $[Q] \in \hat{H}^{-1}(C_p, M)$ inside $\hat{H}^0(C_p, E_\varpi[\lambda])$; this suggests a general principle — a Galois orbit with zero trace still carries detectable residual cohomology — that could substitute for Kolyvagin-style derivative constructions in families where the obstruction to rank-one statements is a vanishing t
A numerical test of the mechanism beyond the paper: for small primes $p \equiv 8 \pmod 9$ with admissible $q$, one could compute the reduced boundary $[(p+1)/3]\bar{Q}_q$ over the residue field and verify the sign prediction (6.16)–(6.17); the worked pairs ($q = 13$, $q = 31$ with $p = 17$) are the seeds, and checking several $q$ for $p = 107$ would stress the Frobenius step against a systematic t
If the mechanism is structural, the natural next testbed is the composite cube-sum problem (products of two primes), where the paper notes several local cubic conditions intervene simultaneously: one would predict that each prime factor contributes its own Frobenius-detected boundary term, with the product of the residue symbols controlling the combined boundary.

Original abstract

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv4,7,8\pmod9$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\equiv8\pmod9$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin--Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take $\lambda$-division before taking the trace, where $\lambda=1-\omega$ and $\omega$ is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.

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, 2026-09-23: 1 request, 7,801 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.16 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.29 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv4,7,8\pmod9$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\equiv8\pmod9$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin--Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take $\lambda$-division before taking the trace, where $\lambda=1-\omega$ and $\omega$ is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.

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	We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv4,7,8\pmod9$ is a sum of two rational cubes.	result 0.60	0.94	0.98	asserted 1.00
s1	Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof.	background 0.91	0.05	0.03	asserted 1.00
s2	For the remaining class $p\equiv8\pmod9$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes.	result 0.55	0.95	0.26	asserted 1.00
s3	The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves.	method 0.99	0.64	0.02	asserted 0.99
s4	The Rankin--Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function.	method 0.89	0.64	0.02	asserted 1.00
s5	Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion.	method 0.91	0.18	0.02	asserted 1.00
s6	A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes.	gap 0.51	0.43	0.01	asserted 1.00
s7	Our decisive idea is to take $\lambda$-division before taking the trace, where $\lambda=1-\omega$ and $\omega$ is a primitive cube root of unity.	method 0.67	0.90	0.03	asserted 0.96
s8	We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$.	result 0.40	0.94	0.03	asserted 1.00
s9	The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.	method 0.94	0.85	0.03	asserted 1.00

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

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sources read

arXiv listing abstract v2

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figures and tables
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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

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