# P3606: Longest orbit of an affine unitriangular map over Z/4Z

- ID: `P3606`
- Reference: `affine-unitriangular-z4-five-period`
- Page: https://theoremdb.org/statements/P3606
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: b5a83bd9bdbf7dfdc7134c15b7360f889389e7bc
- Current Markdown: https://api.theoremdb.org/v1/statements/affine-unitriangular-z4-five-period?representation=markdown
- Record maturity: Reviewed problem

## The problem

Determine the largest orbit length of \(0\) under maps \(x\mapsto Ax+b\) on \((\mathbb Z/4\mathbb Z)^5\), where \(A\) is upper triangular with every diagonal entry equal to 1.

### Remarks

- **Remark.** All matrix and vector arithmetic is modulo 4.
- **Remark.** Every allowed map is a permutation because A is invertible.

### What counts as a solution

- Give coefficients attaining the maximum and verify the orbit, with an algebraic or exhaustive upper bound over all 4^15 maps.

## Status

Status unverified. Orders in affine unitriangular groups are classical, though this orbit formulation may admit a short algebraic answer.

## Research packet

### Background and intake notes

Each candidate has a compact matrix and orbit certificate. Swept coefficient ranges remain useful if the full enumeration stalls.

- Embed (A,b) as a unitriangular 6 by 6 matrix over Z/4Z and compute powers with certified modular arithmetic. Quotient conjugacies only when they preserve the distinguished start point 0.
- Trap: the order of the affine permutation can exceed the orbit length of 0. The target asks for the latter.

- Recorded example: An orbit of length 16 is obtained with rows of A equal to 13221, 01102, 00130, 00012, 00001 and b=(0,3,1,3,3).

### Computational notes

- Two hundred thousand seeded random maps were tested by exact iteration and found maximum sampled orbit length 16. Direct replay returned to 0 after 16 distinct states. The state-space bound is 4^5=1024.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `affine-unitriangular-z4-five-period`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.