# P2444: Image size of x^3+x over prime fields

- ID: `P2444`
- Reference: `cubic-map-image-size`
- Page: https://theoremdb.org/statements/P2444
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/cubic-map-image-size?representation=markdown
- Record maturity: Prospecting problem with recorded work

## The problem

For every prime \(p\ge 5\), is the image of \(x\mapsto x^3+x\) on \(\mathbb{F}_p\) of size \(\lfloor(2p+1)/3\rfloor\)?

### Definitions

- **Definition.** The image size is the number of distinct residues x^3+x modulo p as x ranges over F_p.
- **Definition.** floor denotes the greatest integer at most its argument.

### What counts as a solution

- Prove the image-size formula for every prime p >= 5, or give a counterexample.

## Status

**Resolution status:** Solution submitted.
**Lean verification:** Not Lean-verified.

For every prime p at least 5, the image has size floor((2p+1)/3). [1](#reference-1)

**Recorded scope.** every prime p at least 5

[Read this exact record and its supporting evidence](/records/cmi-claim-exact-image-size)


### Resolution argument

Let \(V_p=\{x^3+x:x\in\mathbf F_p\}\), and let \(\chi=\left(\frac{-3}{p}\right)\). The fiber calculation in the companion record gives
\[
|V_p|=\frac{2p+\chi}{3}.
\]
Quadratic reciprocity gives
\[
\left(\frac{-3}{p}\right)=\left(\frac{p}{3}\right)=
\begin{cases}1,&p\equiv1\pmod3,\\-1,&p\equiv2\pmod3.\end{cases}
\]
Consequently
\[
|V_p|=
\begin{cases}(2p+1)/3,&p\equiv1\pmod3,\\(2p-1)/3,&p\equiv2\pmod3,
\end{cases}
=\left\lfloor\frac{2p+1}{3}\right\rfloor.
\]
This proves the candidate for every requested prime. The result is also the case \(a=0,b=1,c=0\) of the classical formula for the value set of \(x^3+ax^2+bx+c\).

## Research packet

### Saved solution and evidence

**Proposition 1 (The image of x cubed plus x has the proposed size).** For every prime p at least 5, the image has size floor((2p+1)/3).

Let \(V_p=\{x^3+x:x\in\mathbf F_p\}\), and let \(\chi=\left(\frac{-3}{p}\right)\). The fiber calculation in the companion record gives
\[
|V_p|=\frac{2p+\chi}{3}.
\]
Quadratic reciprocity gives
\[
\left(\frac{-3}{p}\right)=\left(\frac{p}{3}\right)=
\begin{cases}1,&p\equiv1\pmod3,\\-1,&p\equiv2\pmod3.\end{cases}
\]
Consequently
\[
|V_p|=
\begin{cases}(2p+1)/3,&p\equiv1\pmod3,\\(2p-1)/3,&p\equiv2\pmod3,
\end{cases}
=\left\lfloor\frac{2p+1}{3}\right\rfloor.
\]
This proves the candidate for every requested prime. The result is also the case \(a=0,b=1,c=0\) of the classical formula for the value set of \(x^3+ax^2+bx+c\).

### Background and intake notes

A useful route record should state how the repeated-root fibers and the character sum are separated. Those are the two places where a quick three-to-one heuristic breaks.

- Original intake status: Novelty and status remain unverified. Value sets of low-degree polynomials are classical and this formula may be standard.
- Counting fibers through the cubic discriminant is the attractive route.
- The discriminant count has exceptional double-root fibers and a quadratic-character sum. Treating every nonzero discriminant uniformly produces the right main term and the wrong constant.

- Recorded example: The image sizes at p=5, 7, 11, 13 are 3, 5, 7, 9.

### Other known results

- **Proposition 2** (supported): Exactly (p-1)/2 values have one preimage, 1+chi values have two, and (p-3-4chi)/6 have three. [1](#reference-1)
- **Proposition 3** (supported): The general four-case formula specializes immediately to the candidate. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (executable material): Direct enumeration confirms the formula for every prime below 50,000. [1](#reference-1)

### Computational notes

- Direct enumeration was performed for all 1227 primes with 5 <= p < 10000. Every image size equaled floor((2p+1)/3).

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `cubic-map-image-size`, 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.

## Lean verification

A solution has been submitted and awaits independent review.

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## References

1. <a id="reference-1"></a>Thomas Brazelton, Joshua Harrington, Matthew Litman, and Tony W. H. Wong, “Residue sums of Dickson polynomials over finite fields,” Journal of Number Theory 264 (2024), 1–26. Published article, pp. 1–2; arXiv v2, p. 1. https://doi.org/10.1016/j.jnt.2024.04.016
   - Also cited at Brazelton, Harrington, Litman, and Wong, Residue sums of Dickson polynomials over finite fields, Journal of Number Theory 264 (2024), page 2 of the published article and page 1 of arXiv:2103.09119v2, displayed cubic value-set formula; DOI 10.1016/j.jnt.2024.04.016
   - Also cited at Brazelton, Harrington, Litman, and Wong, Journal of Number Theory 264 (2024), pages 1-2, cubic formula and historical attribution; references [Kan15] and [vS08]
   - Also cited at Inline Python source below, independently executed on 2026-07-24
   - journal_article; primary source; arXiv:2103.09119v2; checked 2026-07-28
   - Open copy: https://arxiv.org/abs/2103.09119v2
   - Source use: citation_only
   - Records the classical value-set formula for the cubic map used in the packet’s independent proof.
   - Source named by the research packet.