[#P2842] Improve the upper exponent for the longest Pierce remainder chain

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 \(N\to\infty\).

A flat mathematical diagram showing a decreasing chain of integer remainders. — A schematic view of a decreasing chain of integer remainders.

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Definitions and notation

1Context

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.

2Problem setup

Definition 1 (The least nonnegative residue of \(N\) modulo \(a_j\). 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 2 (The chain length). The chain length is the number of positive entries \(a_0,a_1,\ldots\) before the first zero.

Remark 1. The notation \(O_\varepsilon\) allows the implied constant to depend on \(\varepsilon\), but not on \(N\).

3What 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.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (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 \(N\ge2\).^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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 \(N\ge2\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For pierce expansion upper exponent, pins the dated research frontier: 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 \(\varepsilon>0\). Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details 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 \(N\ge2\). Open full record →
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 \(N\ge2\).	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Improve the upper exponent for the longest Pierce remainder chain, record pierce-expansion-upper-exponent-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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 \(N\ge2\). Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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 \(N\ge2\). Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion material

Original intake 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.

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 1. For \(N=10\) and \(a=6\), the positive chain is \(6,4,2\), followed by zero.

Recorded example 2. For \(a\mid N\), the chain has one positive term because the next residue is zero.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemImprove the upper exponent for the longest Pierce remainder chain

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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“Improve the upper exponent for the longest Pierce remainder chain.” TheoremDB. P2842. Problem statement; statement text SHA-256 afb0e87b6cf7723a936f814a0833f0c7445096b218e36f84d4ad00a3dc7643c3. https://theoremdb.org/statement/?ref=P2842

BibTeX

@misc{theoremdb-problem-afb0e87b6cf7723a936f814a0833f0c7445096b218e36f84d4ad00a3dc7643c3,
  title = {{Improve the upper exponent for the longest Pierce remainder chain}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 afb0e87b6cf7723a936f814a0833f0c7445096b218e36f84d4ad00a3dc7643c3},
  url = {https://theoremdb.org/statement/?ref=P2842}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1Lean verification

1References

Packet source. 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. ↗forum · discovery source · checked 2026-07-31Source use: original summary.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.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.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.
Zachary Chase and Mayank Pandey, “On the length of Pierce expansions”. arXiv:2211.08374 (2022). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:2211.08374, checked 2026-07-31 · checked 2026-07-31Source use: original summary.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.Also cited at Full preprint relevant to Improve the upper exponent for the longest Pierce remainder chain.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.

Original CC0 textbook restatement.

[#P2842] Improve the upper exponent for the longest Pierce remainder chain

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