# P2442: Factor degrees in iterates of x^2+x+1 over F_2

- ID: `P2442`
- Reference: `iterated-quadratic-factor-degree`
- Page: https://theoremdb.org/statements/P2442
- 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/iterated-quadratic-factor-degree?representation=markdown
- Record maturity: Prospecting problem with recorded work

## The problem

Let \(f(x)=x^2+x+1\in\mathbb{F}_2[x]\), and let \(f^{\circ n}\) be its \(n\)-fold compositional iterate. Does every irreducible factor of \(f^{\circ n}\) have degree \(2^{\lceil\log_2(n+1)\rceil}\) for every \(n\ge 1\)?

### Problem setup

- **Definition.** f^[1]=f and f^[n+1]=f composed with f^[n].
- **Remark.** Factor degrees and irreducibility are over the field F_2.

### What counts as a solution

- Prove the degree formula for every n, or give an iterate containing a factor of another degree.

## Status

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

For d=2^ceil(log_2(n+1)), the polynomial f^[n] is the product of 2^n/d distinct irreducible polynomials of degree d. [2](#reference-2)

**Recorded scope.** every compositional iterate f^[n] of f(x)=x^2+x+1 over F_2, for every integer n at least 1

[Read this exact record and its supporting evidence](/records/iqfd-claim-exact-factor-degrees)


### Resolution argument

Let \(\sigma\) be Frobenius on \(\overline{\mathbb F}_2\), so \(\sigma(a)=a^2\), and put \(D=\sigma+1\). Since \(f(a)=D(a)+1\) and \(D(1)=0\), induction on composition gives
\[
f^{[n]}(a)=D^n(a)+1.
\]
Set \(d=2^{\lceil\log_2(n+1)\rceil}\). Equivalently, \(d\) is the least power of two greater than \(n\). If \(\alpha\) is a root of \(f^{[n]}\), then \(D^n(\alpha)=1\). Apply \(D^{d-n}\). The exponent is positive, so \(D^{d-n}(1)=0\), and therefore \(D^d(\alpha)=0\). Because \(d\) is a power of two, the binomial identity in characteristic two gives
\[
D^d=(\sigma+1)^d=\sigma^d+1.
\]
Hence \(\alpha^{2^d}=\alpha\), placing every root in \(\mathbb F_{2^d}\).

Let \(m=[\mathbb F_2(\alpha):\mathbb F_2]\). Then \(m\mid d\). If \(m<d\), the power-of-two condition forces \(m\mid d/2\), so \(\alpha\in\mathbb F_{2^{d/2}}\). This would give \(D^{d/2}(\alpha)=\alpha^{2^{d/2}}+\alpha=0\). The minimality of \(d\) gives \(d/2\leq n\), and thus \(D^n(\alpha)=0\), contradicting \(D^n(\alpha)=1\). Every root consequently has degree \(d\) over \(\mathbb F_2\).

The coefficient of \(x\) in the linearized polynomial \(D^n(x)\) is 1, so the derivative of \(f^{[n]}\) is 1. Its \(2^n\) roots are distinct. It follows that there are exactly \(2^n/d\) irreducible factors, each with multiplicity one and degree \(d\). Any root generates \(\mathbb F_{2^d}\), which is the splitting field.

Ore developed the symbolic-composition framework for linearized polynomials. The argument above specializes it and proves the candidate formula. A focused search located the third-iterate factorization in Ahmadi and Monsef-Shokri, while the exact formula for every iterate was not located. Its novelty therefore remains unverified.

## Research packet

### Saved solution and evidence

**Proposition 1 (Every factor has the predicted degree).** For d=2^ceil(log_2(n+1)), the polynomial f^[n] is the product of 2^n/d distinct irreducible polynomials of degree d.

Let \(\sigma\) be Frobenius on \(\overline{\mathbb F}_2\), so \(\sigma(a)=a^2\), and put \(D=\sigma+1\). Since \(f(a)=D(a)+1\) and \(D(1)=0\), induction on composition gives
\[
f^{[n]}(a)=D^n(a)+1.
\]
Set \(d=2^{\lceil\log_2(n+1)\rceil}\). Equivalently, \(d\) is the least power of two greater than \(n\). If \(\alpha\) is a root of \(f^{[n]}\), then \(D^n(\alpha)=1\). Apply \(D^{d-n}\). The exponent is positive, so \(D^{d-n}(1)=0\), and therefore \(D^d(\alpha)=0\). Because \(d\) is a power of two, the binomial identity in characteristic two gives
\[
D^d=(\sigma+1)^d=\sigma^d+1.
\]
Hence \(\alpha^{2^d}=\alpha\), placing every root in \(\mathbb F_{2^d}\).

Let \(m=[\mathbb F_2(\alpha):\mathbb F_2]\). Then \(m\mid d\). If \(m<d\), the power-of-two condition forces \(m\mid d/2\), so \(\alpha\in\mathbb F_{2^{d/2}}\). This would give \(D^{d/2}(\alpha)=\alpha^{2^{d/2}}+\alpha=0\). The minimality of \(d\) gives \(d/2\leq n\), and thus \(D^n(\alpha)=0\), contradicting \(D^n(\alpha)=1\). Every root consequently has degree \(d\) over \(\mathbb F_2\).

The coefficient of \(x\) in the linearized polynomial \(D^n(x)\) is 1, so the derivative of \(f^{[n]}\) is 1. Its \(2^n\) roots are distinct. It follows that there are exactly \(2^n/d\) irreducible factors, each with multiplicity one and degree \(d\). Any root generates \(\mathbb F_{2^d}\), which is the splitting field.

Ore developed the symbolic-composition framework for linearized polynomials. The argument above specializes it and proves the candidate formula. A focused search located the third-iterate factorization in Ahmadi and Monsef-Shokri, while the exact formula for every iterate was not located. Its novelty therefore remains unverified.

### Background and intake notes

The trap is a composition induction that tracks degrees and drops the trace state governing the next split. Reusable attempts should record the auxiliary trace invariant used for every factor orbit.

- Original intake status: Novelty and status remain unverified. Stability and factorization of polynomial iterates are established topics.
- Induction through composition is the attractive route: f^[n+1]=(f^[n])^2+f^[n]+1.
- Irreducibility under composition depends on trace data inside every current factor. The number of factors doubles across some steps, so checking a single representative loses information.

- Recorded example: The common factor degrees for n=1,...,9 are 2, 4, 4, 8, 8, 8, 8, 16, 16.

### Other known results

- **Proposition 2** (supported): Ahmadi and Monsef-Shokri give a factorization of f^[3] into two quartics while proving instability of binary trinomials. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (computational evidence): Full factorization over F_2 through n=12 gives only the predicted squarefree factor degrees and counts.

### Computational notes

- Exact factorization over F_2 was completed for 1 <= n <= 9, whose iterate degrees reach 512. Every factor was squarefree and had the predicted common degree.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `iterated-quadratic-factor-degree`, 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>Reference record for “iterated-quadratic-factor-degree,” arxiv.org, checked 2026-08-01. Packet-wide source named in the dataset provenance. https://arxiv.org/abs/1810.03142
   - Also cited at Ahmadi and Monsef-Shokri, A note on the stability of trinomials over finite fields, Theorem 7 and its n=1, s=1 case on page 6; Finite Fields and Their Applications 63 (2020), article 101649, DOI 10.1016/j.ffa.2020.101649
   - preprint; reference source; arXiv:1810.03142, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Supplies the source context used to assemble the packet for “iterated quadratic factor degree.”
   - Source named by the research packet.
2. <a id="reference-2"></a>Reference record for “Every factor has the predicted degree,” arxiv.org, checked 2026-08-01. Wu and Liu, Linearized polynomials over finite fields revisited, equations (1)-(2), section 2.1, and Theorems 2.1-2.2 for the Frobenius and composition algebra; the specialized all-n proof is complete in this record https://arxiv.org/abs/1211.5475
   - preprint; reference source; arXiv:1211.5475, checked 2026-08-01; checked 2026-08-01
   - Source use: citation_only
   - Documents the claim, method, computation, or status recorded as “Every factor has the predicted degree” in this packet.