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

- ID: `P2850`
- Reference: `unbounded-integral-points-minimal-elliptic-curves`
- Page: https://theoremdb.org/statements/P2850
- Record maturity: Reviewed problem with recorded work

## 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 \(\mathbb Q\).

### Problem setup

- **Definition.** A global minimal integral Weierstrass equation is an integral Weierstrass model whose discriminant has minimal valuation at every prime.
- **Definition.** 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.** The number \(N(E)\) is independent of the choice among globally minimal integral equations because allowed integral changes of minimal coordinates preserve integral points.

### What 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.

## Status

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](#reference-3) [2](#reference-2) [4](#reference-4) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- 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 \(N(E)\).

- Recorded example: 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `unbounded-integral-points-minimal-elliptic-curves`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>MathOverflow question 50661, “Unbounded numbers of integral points on global minimal elliptic curves,” checked 2026-08-01. Original CC0 dichotomy statement written after reviewing every answer and comment and later work on moments and record curves. https://mathoverflow.net/questions/50661/unboundedness-of-number-of-integral-points-on-elliptic-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.
   - forum; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Noam D. Elkies, “Three lectures on elliptic surfaces and curves of high rank,” arXiv:0709.2908 (2007). abstract and lectures on Mordell-Weil rank records and elliptic-surface construction methods https://arxiv.org/abs/0709.2908
   - preprint; reference source; arXiv:0709.2908, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>Levent Alpöge and Wei Ho, “The second moment of the number of integral points on elliptic curves is bounded,” arXiv:1807.03761 (2018). abstract and bounded-second-moment theorem for numbers of S-integral points in height-ordered elliptic-curve families https://arxiv.org/abs/1807.03761
   - preprint; reference source; arXiv:1807.03761, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - 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.
4. <a id="reference-4"></a>Navvye Anand, “On Bounds and Diophantine Properties of Elliptic Curves,” arXiv:2407.09558 (2024). abstract and explicit bounds and families for integral points on Mordell curves https://arxiv.org/abs/2407.09558
   - preprint; reference source; arXiv:2407.09558, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - 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.