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

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

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 \(N\to\infty\).

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.

Definition 3 (The notation \(O_\varepsilon\) allows the implied constant to depend on \(\varepsilon\), but not on \(N\). The notation \(O_\varepsilon\) allows the implied constant to depend on \(\varepsilon\), but not on \(N\).

Remark 1. 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.

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

Current 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]

1Records

Record	Kind	Assessment
	Result	Supported
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 Defines the dated frontier and the exact remainder that research on this problem must resolve. 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 →
	Trace	Reported
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 materialContext, examples, and computations

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 connectTyped relations and evidence flow

ProblemImprove the upper exponent for the longest Pierce remainder chain

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Improve the upper exponent for the longest Pierce remainder chain,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/pierce-expansion-upper-exponent

This page as plain text: pierce-expansion-upper-exponent.md

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

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

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance