# P2840: A two-variable polynomial whose integer image is exactly the nonnegative integers

- ID: `P2840`
- Reference: `polynomial-image-z2-nonnegative-integers`
- Page: https://theoremdb.org/statements/P2840
- Record maturity: Reviewed problem with recorded work

## Problem

Does there exist a polynomial \(F\in\mathbb Q[X,Y]\) whose image on the full integer lattice is exactly \(\mathbb Z_{\ge 0}\)? Thus \(F(x,y)\) must be a nonnegative integer for every \((x,y)\in\mathbb Z^2\), and every nonnegative integer must equal \(F(x,y)\) for at least one integer pair.

### Problem setup

- **Definition.** The image \(F(\mathbb Z^2)\) is the set \(\{F(x,y):x,y\in\mathbb Z\}\).
- **Remark.** A rational-coefficient polynomial may be integer-valued on \(\mathbb Z^2\) even when some coefficients are not integers.
- **Definition.** Exact image means both containment \(F(\mathbb Z^2)\subseteq\mathbb Z_{\ge0}\) and surjectivity onto every member of \(\mathbb Z_{\ge0}\).

### What counts as a solution

- Give an explicit \(F\in\mathbb Q[X,Y]\) and prove the equality \(F(\mathbb Z^2)=\mathbb Z_{\ge0}\), including integrality and nonnegativity on every integer pair.
- Alternatively, prove unconditionally that no such polynomial exists. A conclusion conditional on Vojta's conjecture does not meet acceptance.

## Status

UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem. Give an explicit \(F\in\mathbb Q[X,Y]\) and prove the equality \(F(\mathbb Z^2)=\mathbb Z_{\ge0}\), including integrality and nonnegativity on every integer pair. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem. Give an explicit \(F\in\mathbb Q[X,Y]\) and prove the equality \(F(\mathbb Z^2)=\mathbb Z_{\ge0}\), including integrality and nonnegativity on every integer pair.

UNKNOWN as of 2026-07-31. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem.

A complete resolution must satisfy this condition: Give an explicit \(F\in\mathbb Q[X,Y]\) and prove the equality \(F(\mathbb Z^2)=\mathbb Z_{\ge0}\), including integrality and nonnegativity on every integer pair.

### Background and intake notes

The question asks whether two unrestricted integer parameters admit a single polynomial coding of the nonnegative integers without stray values. Degree bounds, positivity certificates, and exhaustive searches through normalized low-degree families can be retained as intermediate artifacts.

- Original intake status: UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem.
- On 2026-07-27 the full MathOverflow thread, its eight answers, and answer comments were checked. Proposed expressions either use restricted domains, extra variables, nonpolynomial operations, or leave one of the two image inclusions unproved.
- Rawson, arXiv:2403.09440, gives a negative answer conditional on Vojta's conjecture. The conjectural hypothesis prevents that paper from settling this record.
- Classical pairing polynomials cover \(\mathbb Z_{\ge0}^2\), while this problem uses all of \(\mathbb Z^2\). Extending the domain can introduce negative or nonintegral values.
- Trap: representing each nonnegative integer is only half of the target. Every integer input pair must also produce a nonnegative integer, and no positive surjectivity argument may silently restrict the signs of \(x\) and \(y\).

- Recorded example: The Cantor polynomial \((X+Y)(X+Y+1)/2+Y\) pairs nonnegative integer inputs, but it takes unwanted values when \(X\) and \(Y\) range over all integers.
- Recorded example: The polynomial \(X^2+Y^2\) is always nonnegative on integer pairs, but its image omits many nonnegative integers.

### Open directions

- **Route 1** (reported): Give an explicit \(F\in\mathbb Q[X,Y]\) and prove the equality \(F(\mathbb Z^2)=\mathbb Z_{\ge0}\), including integrality and nonnegativity on every integer pair. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `polynomial-image-z2-nonnegative-integers`, 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>Polynomial representing all nonnegative integers, MathOverflow question 9731. Original CC0 restatement written after reviewing the question, all eight answers, their comments, and later conditional work. mathoverflow.net checked 2026-08-01. Original CC0 restatement written after reviewing the question, all eight answers, their comments, and later conditional work. https://mathoverflow.net/questions/9731/polynomial-representing-all-nonnegative-integers
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For A two-variable polynomial whose integer image is exactly the nonnegative integers: UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem.
   - Source named by the research packet.
2. <a id="reference-2"></a>James Rawson, “On a problem posed by Bjorn Poonen”. arXiv:2403.09440 (2024). Full preprint relevant to A two-variable polynomial whose integer image is exactly the nonnegative integers. https://arxiv.org/abs/2403.09440
   - preprint; reference source; arXiv:2403.09440, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For A two-variable polynomial whose integer image is exactly the nonnegative integers: UNKNOWN as of 2026-07-27. The MathOverflow thread has no accepted construction. Rawson gives a conditional negative result using Vojta's conjecture, and the dated search found no unconditional existence or nonexistence theorem.