# P2842: Improve the upper exponent for the longest Pierce remainder chain

- ID: `P2842`
- Reference: `pierce-expansion-upper-exponent`
- Page: https://theoremdb.org/statements/P2842
- Record maturity: Reviewed problem with recorded work

## Problem

For integers \(N\ge2\) and \(1\le a\le N\), set \(a_0=a\) and \(a_{j+1}\) equal to the least nonnegative residue of \(N\) modulo \(a_j\) until the first zero occurs. Let \(L(N)\) be the maximum number of positive terms over all starting values \(a\). Prove that there is a constant \(\delta>2/177\) such that, for every \(\varepsilon>0\), \(L(N)=O_\varepsilon(N^{1/3-\delta+\varepsilon})\) as \(N\to\infty\).

### Problem setup

- **Definition.** The least nonnegative residue of \(N\) modulo \(a_j\) is the unique integer \(r\) with \(0\le r<a_j\) and \(N\equiv r\pmod{a_j}\).
- **Definition.** The chain length is the number of positive entries \(a_0,a_1,\ldots\) before the first zero.
- **Remark.** The notation \(O_\varepsilon\) allows the implied constant to depend on \(\varepsilon\), but not on \(N\).

### What counts as a solution

- Exhibit a fixed \(\delta>2/177\) and prove that for every \(\varepsilon>0\) there is a constant \(C_\varepsilon\) with \(L(N)\le C_\varepsilon N^{1/3-\delta+\varepsilon}\) for every \(N\ge2\).
- State all exceptional finite ranges and parameter dependencies explicitly. Computation may discharge a finite range after the uniform argument supplies a concrete cutoff.

## Status

UNKNOWN as of 2026-07-31. Chase and Pandey prove \(L(N)=O_\varepsilon(N^{1/3-2/177+\varepsilon})\). The dated search did not find a larger published saving, so this record asks for a strict improvement over that theorem. Exhibit a fixed \(\delta>2/177\) and prove that for every \(\varepsilon>0\) there is a constant \(C_\varepsilon\) with \(L(N)\le C_\varepsilon N^{1/3-\delta+\varepsilon}\) for every \(N\ge2\). [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** UNKNOWN as of 2026-07-31. Chase and Pandey prove \(L(N)=O_\varepsilon(N^{1/3-2/177+\varepsilon})\). The dated search did not find a larger published saving, so this record asks for a strict improvement over that theorem. Exhibit a fixed \(\delta>2/177\) and prove that for every \(\varepsilon>0\) there is a constant \(C_\varepsilon\) with \(L(N)\le C_\varepsilon N^{1/3-\delta+\varepsilon}\) for every \(N\ge2\).

UNKNOWN as of 2026-07-31. Chase and Pandey prove \(L(N)=O_\varepsilon(N^{1/3-2/177+\varepsilon})\). The dated search did not find a larger published saving, so this record asks for a strict improvement over that theorem.

A complete resolution must satisfy this condition: Exhibit a fixed \(\delta>2/177\) and prove that for every \(\varepsilon>0\) there is a constant \(C_\varepsilon\) with \(L(N)\le C_\varepsilon N^{1/3-\delta+\varepsilon}\) for every \(N\ge2\).

### Background and intake notes

The target advances a published quantitative frontier by any fixed exponent. Exact remainder-chain tables and counts for the bad transition patterns remain useful when a later proof changes the exponent again.

- Original intake status: UNKNOWN as of 2026-07-27. Chase and Pandey prove \(L(N)=O_\varepsilon(N^{1/3-2/177+\varepsilon})\). The dated search did not find a larger published saving, so this record asks for a strict improvement over that theorem.
- On 2026-07-27 the MathOverflow question, its answer, and comments were checked. The thread predates the current upper bound and does not itself resolve the sharper target stated here.
- Chase and Pandey, arXiv:2211.08374, improve the former \(1/3+\varepsilon\) exponent to \(1/3-2/177+\varepsilon\). The requested \(\delta>2/177\) begins strictly beyond their theorem.
- The recurrence is monotone while positive because \(0\le a_{j+1}<a_j\), so every chain terminates. This elementary fact fixes the length convention and supports exact computation.
- Reusable search data include maximizing starting values, full remainder chains, dyadic transition counts, and certificates for any decomposition used in an upper-bound proof.
- Trap: a better estimate on average over \(N\), an improvement for almost all \(N\), or a smaller hidden constant does not improve the worst-case exponent required here.

- Recorded example: For \(N=10\) and \(a=6\), the positive chain is \(6,4,2\), followed by zero.
- Recorded example: For \(a\mid N\), the chain has one positive term because the next residue is zero.

### Open directions

- **Route 1** (reported): Exhibit a fixed \(\delta>2/177\) and prove that for every \(\varepsilon>0\) there is a constant \(C_\varepsilon\) with \(L(N)\le C_\varepsilon N^{1/3-\delta+\varepsilon}\) for every \(N\ge2\). [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `pierce-expansion-upper-exponent`, 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>Improving known bounds for Pierce expansions, MathOverflow question 164129. Original CC0 bound-improvement target written after reviewing the question, its answer and comments, and Chase and Pandey's later theorem. mathoverflow.net checked 2026-08-01. Original CC0 bound-improvement target written after reviewing the question, its answer and comments, and Chase and Pandey's later theorem. https://mathoverflow.net/questions/164129/improving-known-bounds-for-pierce-expansions-cash-prize
   - 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 Improve the upper exponent for the longest Pierce remainder chain: UNKNOWN as of 2026-07-27. Chase and Pandey prove \(L(N)=O_\varepsilon(N^{1/3-2/177+\varepsilon})\). The dated search did not find a larger published saving, so this record asks for a strict improvement over that theorem.
   - Source named by the research packet.
2. <a id="reference-2"></a>Zachary Chase and Mayank Pandey, “On the length of Pierce expansions”. arXiv:2211.08374 (2022). Full preprint relevant to Improve the upper exponent for the longest Pierce remainder chain. https://arxiv.org/abs/2211.08374
   - preprint; reference source; arXiv:2211.08374, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - Source used to assess the problem's recorded status.
   - For Improve the upper exponent for the longest Pierce remainder chain: UNKNOWN as of 2026-07-27. Chase and Pandey prove \(L(N)=O_\varepsilon(N^{1/3-2/177+\varepsilon})\). The dated search did not find a larger published saving, so this record asks for a strict improvement over that theorem.