[#P2850] Unbounded numbers of integral points on global minimal elliptic curves

A mathematical schematic of Unbounded numbers of integral points on global minimal elliptic curves. — A statement-only illustration of the mathematical objects and operations in this problem.

Problem. For an elliptic curve \(E/\mathbb Q\), choose a global minimal integral Weierstrass equation and let \(N(E)\) be the number of affine integral solutions \((x,y)\in\mathbb Z^2\) on that equation. Prove that \(\sup_E N(E)=\infty\), or prove instead that one absolute constant bounds \(N(E)\) for every elliptic curve over \(\mathbb Q\).

1Context

The problem separates abundant integral points caused by coordinate scaling from those intrinsic to minimal models. Databases of certified minimal curves and distinct integral points provide durable lower bounds.

2Problem setup

Definition 1. A global minimal integral Weierstrass equation is an integral Weierstrass model whose discriminant has minimal valuation at every prime.

Definition 2. An affine integral point is a pair \((x,y)\in\mathbb Z^2\) satisfying the chosen Weierstrass equation; the point at infinity is not counted.

Remark 1. The number \(N(E)\) is independent of the choice among globally minimal integral equations because allowed integral changes of minimal coordinates preserve integral points.

3What counts as a solution

For unboundedness, construct or prove the existence of elliptic curves \(E_r/\mathbb Q\) with certified global minimal equations and \(N(E_r)\to\infty\).
For boundedness, give an absolute constant \(C\) and prove \(N(E)\le C\) for every elliptic curve over \(\mathbb Q\) on its global minimal integral model.

1Status

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: Large explicit examples and average bounds are known, but the checked sources do not prove unboundedness across global minimal models or an absolute uniform bound. Exact unresolved remainder: For unboundedness, construct or prove the existence of elliptic curves \(E_r/\mathbb Q\) with certified global minimal equations and \(N(E_r)\to\infty\). For boundedness, give an absolute constant \(C\) and prove \(N(E)\le C\) for every elliptic curve over \(\mathbb Q\) on its global minimal integral model.^[3][2][4][1]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: Large explicit examples and average bounds are known, but the checked sources do not prove unboundedness across global minimal models or an absolute uniform bound. Exact unresolved remainder: For unboundedness, construct or prove the existence of elliptic curves \(E_r/\mathbb Q\) with certified global minimal equations and \(N(E_r)\to\infty\). For boundedness, give an absolute constant \(C\) and prove \(N(E)\le C\) for every elliptic curve over \(\mathbb Q\) on its global minimal integral model.^[3][2][4][1] Relevance to this problem For Unbounded numbers of integral points on global minimal elliptic curves, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 3, Reference 2, Reference 4, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: Large explicit examples and average bounds are known, but the checked sources do not prove unboundedness across global minimal models or an absolute uniform bound. Exact unresolved remainder: For unboundedness, construct or prove the existence of elliptic curves \(E_r/\mathbb Q\) with certified global minimal equations and \(N(E_r)\to\infty\). For boundedness, give an absolute constant \(C\) and prove \(N(E)\le C\) for every elliptic curve over \(\mathbb Q\) on its global minimal integral model. Open full record →
	Trace	Reported
attempt · Route 1 For unboundedness, construct or prove the existence of elliptic curves \(E_r/\mathbb Q\) with certified global minimal equations and \(N(E_r)\to\infty\). For boundedness, give an absolute constant \(C\) and prove \(N(E)\le C\) for every elliptic curve over \(\mathbb Q\) on its global minimal integral model.^[1] Relevance to this problem For Unbounded numbers of integral points on global minimal elliptic curves, record unbounded-integral-points-minimal-elliptic-curves-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: For unboundedness, construct or prove the existence of elliptic curves \(E_r/\mathbb Q\) with certified global minimal equations and \(N(E_r)\to\infty\). Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. For unboundedness, construct or prove the existence of elliptic curves \(E_r/\mathbb Q\) with certified global minimal equations and \(N(E_r)\to\infty\). For boundedness, give an absolute constant \(C\) and prove \(N(E)\le C\) for every elliptic curve over \(\mathbb Q\) on its global minimal integral model. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-27. Large explicit examples and average bounds are known, but the checked sources do not prove unboundedness across global minimal models or an absolute uniform bound.

On 2026-07-27 all MathOverflow answers and comments were checked. They distinguish large examples on minimal models from changes of variables that manufacture integral points on nonminimal equations.
Elkies, arXiv:0709.2908, records elliptic curves with thousands of integral-point pairs and discusses the minimal-model issue. Any new record is finite evidence rather than a proof of unboundedness.
Alpöge and Ho, arXiv:1807.03761, prove average moment bounds in families. Average control permits exceptional curves and therefore does not supply a uniform absolute bound.
A search also checked recent work on integral points and binary cubic forms, including arXiv:2407.09558. The audit did not verify that its updated version resolves the minimal-model dichotomy, so specialist reconciliation is required.
Trap: scaling an equation can turn rational points into integral coordinate pairs while destroying global minimality. Every lower-bound family must certify the model used to define \(N(E)\).
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. A pair \((x,y)\) and \((x,-y-a_1x-a_3)\) usually give two integral points on a general Weierstrass equation, so counting conventions should retain both affine solutions when distinct.

Computational notes

Published record curves give substantial finite lower bounds for \(\sup_E N(E)\). Each entry needs a minimality certificate and exact point verification before reuse.

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Unbounded numbers of integral points on global minimal elliptic curves,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/unbounded-integral-points-minimal-elliptic-curves

This page as plain text: unbounded-integral-points-minimal-elliptic-curves.md

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 50661, “Unbounded numbers of integral points on global minimal elliptic curves,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Original CC0 dichotomy statement written after reviewing every answer and comment and later work on moments and record curves.Also cited at Full question, answers, and visible comments concerning Unbounded numbers of integral points on global minimal elliptic curves; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Unbounded numbers of integral points on global minimal elliptic curves, the reviewed source scope is Full question, answers, and visible comments concerning Unbounded numbers of integral points on global minimal elliptic curves; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
arXiv preprint 0709.2908, linked primary source for “Unbounded numbers of integral points on global minimal elliptic curves,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:0709.2908, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and lectures on Mordell-Weil rank records and elliptic-surface construction methods.Source used to assess the problem's recorded status.For Unbounded numbers of integral points on global minimal elliptic curves, this source was excluded as status evidence because high rank alone neither counts integral points nor verifies global minimal models; it is retained to document source-review history.
Levent Alpöge and Wei Ho, “The second moment of the number of integral points on elliptic curves is bounded,” arXiv:1807.03761 (2018). Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:1807.03761, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and bounded-second-moment theorem for numbers of S-integral points in height-ordered elliptic-curve families.Source used to assess the problem's recorded status.For Unbounded numbers of integral points on global minimal elliptic curves, this source gives strong average bounds while allowing exceptions, so it proves neither a uniform bound nor unboundedness on global minimal models.
arXiv preprint 2407.09558, linked primary source for “Unbounded numbers of integral points on global minimal elliptic curves,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:2407.09558, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and explicit bounds and families for integral points on Mordell curves.Source used to assess the problem's recorded status.For Unbounded numbers of integral points on global minimal elliptic curves, this source supplies nearby examples and bounds; the packet must separately verify global minimal equations and cannot infer unboundedness from this source.

Original CC0 textbook restatement.

[#P2850] Unbounded numbers of integral points on global minimal elliptic curves

1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance