[#P2662] Most irreducible constant slices of a sparse quintic over F_101

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Problem. For nonzero \(a,b\in\mathbb F_{101}\), let \(N(a,b)\) be the number of \(c\in\mathbb F_{101}\) for which \(x^5+a x^2+b x+c\) is irreducible over \(\mathbb F_{101}\). Determine \(\max_{a,b\ne0}N(a,b)\).

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Definitions and notation

1Context

A slice is a natural independent work unit, and every positive test has a short finite-field certificate.

2Problem setup

Definition 1. Irreducible means having no factor of positive degree below five over F_101.

Remark 1. Each pair (a,b) defines a slice of 101 polynomials indexed by c.

3What counts as a solution

Give coefficients attaining the maximum and an exact sweep or theorem certifying every one of the 10000 slices.

1Status

Saved packet · July 25, 2026

What counts as a solution

Submitted solution (The exact maximum slice count is 29). A complete sweep finds 29 irreducible quintics in each maximizing slice and certifies that every other slice has at most 28.^[3]

Verification

The answer is \[ \max_{a,b\in\mathbb F_{101}^{*}}N(a,b)=29. \] Exactly 100 ordered pairs attain the maximum. They have the compact description \[ a^4b^{-3}=62, \] or equivalently \(a^4=62b^3\), in \(\mathbb F_{101}\). The pair \((a,b)=(1,32)\) is one representative. Its 29 successful constants are \[ \{2,3,7,8,11,12,13,15,16,19,26,29,30,33,38,39,41,43,61,65,72,74,79,83,88,89,96,98,99\}. \] The candidate pair \((95,57)\) belongs to the same maximizing class and has the same displayed constant set.

The characterization follows from an exact sweep together with scaling. For \(u\ne0\), substitution and normalization give \[ u^{-5}f_{a,b,c}(ux)=f_{a u^{-3},b u^{-4},c u^{-5}}(x). \] This preserves irreducibility and leaves \(a^4b^{-3}\) unchanged. The action on nonzero coefficient pairs is free because \(u^3=u^4=1\) forces \(u=1\). Its 100-element orbits are exactly the fibers of \(a^4b^{-3}\). The exhaustive table has one maximizing orbit, with invariant 62.

Across all 10,000 slices, the slice-count histogram is \[ 13{:}200,14{:}400,15{:}300,16{:}700,17{:}600,18{:}1100,19{:}700,20{:}1100,21{:}900,22{:}1100,23{:}600,24{:}900,25{:}600,26{:}400,27{:}200,28{:}100,29{:}100. \] The counts sum to 10,000 slices. Those slices contain 204,200 irreducible polynomials in total.

1Packet records

Show recorded summaries

Record	Kind	Assessment
A complete sweep finds 29 irreducible quintics in each maximizing slice and certifies that every other slice has at most 28.	Result	Evidence: Reproduced A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1 A complete sweep finds 29 irreducible quintics in each maximizing slice and certifies that every other slice has at most 28.^[3] Relevance to this problem For Most irreducible constant slices of a sparse quintic over F_101, record iqf101-claim-exact-maximum-29 (“The exact maximum slice count is 29”) records a bound, answer, status fact, or structural consequence. The record states: A complete sweep finds 29 irreducible quintics in each maximizing slice and certifies that every other slice has at most 28. Evidence ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope. Record state established Scope all polynomials x^5+ax^2+bx+c over F_101 with nonzero a and b Sources Reference 3 Details The answer is \[ \max_{a,b\in\mathbb F_{101}^{*}}N(a,b)=29. \] Exactly 100 ordered pairs attain the maximum. They have the compact description \[ a^4b^{-3}=62, \] or equivalently \(a^4=62b^3\), in \(\mathbb F_{101}\). The pair \((a,b)=(1,32)\) is one representative. Its 29 successful constants are \[ \{2,3,7,8,11,12,13,15,16,19,26,29,30,33,38,39,41,43,61,65,72,74,79,83,88,89,96,98,99\}. \] The candidate pair \((95,57)\) belongs to the same maximizing class and has the same displayed constant set. The characterization follows from an exact sweep together with scaling. For \(u\ne0\), substitution and normalization give \[ u^{-5}f_{a,b,c}(ux)=f_{a u^{-3},b u^{-4},c u^{-5}}(x). \] This preserves irreducibility and leaves \(a^4b^{-3}\) unchanged. The action on nonzero coefficient pairs is free because \(u^3=u^4=1\) forces \(u=1\). Its 100-element orbits are exactly the fibers of \(a^4b^{-3}\). The exhaustive table has one maximizing orbit, with invariant 62. Across all 10,000 slices, the slice-count histogram is \[ 13{:}200,14{:}400,15{:}300,16{:}700,17{:}600,18{:}1100,19{:}700,20{:}1100,21{:}900,22{:}1100,23{:}600,24{:}900,25{:}600,26{:}400,27{:}200,28{:}100,29{:}100. \] The counts sum to 10,000 slices. Those slices contain 204,200 irreducible polynomials in total. Open full record →
Rabin supplies the exact test, and prescribed-coefficient papers place the family in a well-studied class. This audit found no published value for the fixed maximum.	Trace	Outcome: Inconclusive Evidence: Inconclusive The recorded search or audit ended without settling the question.
attempt · Route 1 Rabin supplies the exact test, and prescribed-coefficient papers place the family in a well-studied class. This audit found no published value for the fixed maximum.^[1][3][2] Relevance to this problem For Most irreducible constant slices of a sparse quintic over F_101, record iqf101-attempt-literature-audit (“The literature supplies general methods for prescribed coefficients”) documents a concrete method, search boundary, or failed route. The record states: Rabin supplies the exact test, and prescribed-coefficient papers place the family in a well-studied class. Evidence InconclusiveThe recorded search or audit ended without settling the question. Scope published irreducibility tests and prescribed-coefficient enumeration literature checked for the fixed F_101 quintic slice maximum through 2026-07-25 Sources Reference 1, Reference 3, Reference 2 What happened Rabin's 1980 paper gives the finite-field factorization and irreducibility machinery used in the exhaustive artifact. Granger gives a deterministic method for exact expressions when the first \(\ell<p\) coefficients of a monic degree-\(n\) polynomial are prescribed and \(\gcd(n,p)=1\). Here a slice fixes the first four coefficients after the leading term, namely \(0,0,a,b\), with \(\ell=4\), \(p=101\), and \(n=5\). Summing over the constant term produces \(N(a,b)\). Gao, Kuttner, and Wang develop a general group-algebra generating-function framework for prescribed leading and ending coefficients. Their framework confirms that exact coefficient counts belong to an established research area. A focused search for this sparse degree-five family over \(\mathbb F_{101}\), the value 29, and the invariant class 62 found no published table or theorem stating this fixed maximum. Publication novelty remains unverified. Related: Computation 1 Open full record →
A dependency-free C++17 program tests every polynomial, records every successful constant, and cross-checks two finite-field irreducibility criteria.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 A dependency-free C++17 program tests every polynomial, records every successful constant, and cross-checks two finite-field irreducibility criteria.^[3] Relevance to this problem For Most irreducible constant slices of a sparse quintic over F_101, record iqf101-artifact-rabin-sweep (“Complete Rabin sweep with a second exact criterion”) supplies evidence or a replay used to check the packet. The record states: A dependency-free C++17 program tests every polynomial, records every successful constant, and cross-checks two finite-field irreducibility criteria. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope the 1,010,000 monic quintics x^5+ax^2+bx+c with a,b nonzero and c arbitrary over F_101 Sources Reference 3 Entry point `Join source_lines with newline and append a final newline. Compile with c++ -O3 -std=c++17, run into sweep.tsv, inspect the first three lines, and hash the whole output. Hash tail -n +4 for the canonical table. Hash tail -n +4 piped through cut -f1-3 for the count projection.` Runtime C++17 compiler and standard POSIX text tools Details For a monic degree-five polynomial \(f\) over \(\mathbb F_q\), Rabin's criterion reduces to \[ \gcd(f,x^q-x)=1,\qquad x^{q^5}-x\equiv0\pmod f. \] The program applies these tests with \(q=101\), using exact modular polynomial arithmetic. It also checks every result through \[ \gcd(f,x^{q^2}-x)=1. \] This second condition is equivalent here because every reducible quintic has an irreducible factor of degree one or two, and \(x^{q^2}-x\) contains every monic irreducible whose degree divides two. The program exits if the two decisions differ. They agree on all 1,010,000 inputs. The output begins with three summary lines. It then writes one canonical row for every pair in lexicographic order, formatted as `a<TAB>b<TAB>N<TAB>c_1,c_2,...<LF>`. The 10,000-row table has SHA-256 digest `3f5ed42371a2aaf93ee844d9c059a8b97bf2b218383405627d4daf74fec97807`. Its three-column count projection has digest `dceeaf0af622ab8f01bfa890285653501ec544b8aa34807c1255e3badc79f231`. The complete output, including summary lines, has digest `66e4bb3a3579a47ff36fc2e1cb72ec4c917cb4dd3acea76c85693becc104a123`. Related: Computation 1 Open full record →
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Notes and companion material

Original intake status. SOLVED in the independently reviewed TheoremDB packet as of 2026-08-01. A complete sweep finds 29 irreducible quintics in each maximizing slice and certifies that every other slice has at most 28.

Independent isolated execution completed successfully for Complete Rabin sweep with a second exact criterion. Every embedded assertion passed and the run reproduced the selected exact result: A complete sweep finds 29 irreducible quintics in each maximizing slice and certifies that every other slice has at most 28.
Fresh exact-title, parameter, primary-source, and controlled-corpus searches were completed on 2026-08-01.

Recorded example 1. For (a,b)=(95,57), the 29 successful constants are [2,3,7,8,11,12,13,15,16,19,26,29,30,33,38,39,41,43,61,65,72,74,79,83,88,89,96,98,99].

Computational notes

Exact Rabin tests verified all 29 displayed quintics, giving a lower bound of 29. For fixed a,b, the map x->x^5+a*x^2+b*x has image size at least ceil(101/5)=21. Every c producing an irreducible quintic must avoid the negative of this image, so N(a,b)<=80.

How the 3 records connect

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

ProblemMost irreducible constant slices of a sparse quintic over F_101

All 2 recorded relations between these records and the problem

2See also

Least trinomial multiple of a primitive degree-61 polynomialfinite fields
Rank log-concavity for symmetric binary matrices through order fiftyfinite fields
An APN permutation of the 256-element fieldfinite fields

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Plain text

“Most irreducible constant slices of a sparse quintic over F_101.” TheoremDB. P2662. Problem statement; statement text SHA-256 8564f75c391bb732fc1e603ceeeb07f4ee9fcbed181a79fc1c7140d6f9530590. https://theoremdb.org/statement/?ref=P2662

BibTeX

@misc{theoremdb-problem-8564f75c391bb732fc1e603ceeeb07f4ee9fcbed181a79fc1c7140d6f9530590,
  title = {{Most irreducible constant slices of a sparse quintic over F\_101}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 8564f75c391bb732fc1e603ceeeb07f4ee9fcbed181a79fc1c7140d6f9530590},
  url = {https://theoremdb.org/statement/?ref=P2662}
}

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This problem includes 3 records joined by 2 typed links, sourced from doi.org^[3], current as of July 25, 2026.

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1References

Robert Granger, On the Enumeration of Irreducible Polynomials over GF(q) with Prescribed Coefficients, Finite Fields and Their Applications 57 (2019), 156-229; Michael O. Rabin, SIAM Journal on Computing 9 (1980), 273-280; Zhicheng Gao, Simon Kuttner, and Qiang Wang, Finite Fields and Their Applications 80 (2022), 102023. ↗journal article · primary source · version of record · checked 2026-08-01Source use: original summary.This source fixes the published convention, theorem, formula, or independent answer used to check the packet resolution.For Most irreducible constant slices of a sparse quintic over F_101, the reviewed source scope is Robert Granger, On the Enumeration of Irreducible Polynomials over GF(q) with Prescribed Coefficients, Finite Fields and Their Applications 57 (2019), 156-229; Michael O. Rabin, SIAM Journal on Computing 9 (1980), 273-280; Zhicheng Gao, Simon Kuttner, and Qiang Wang, Finite Fields and Their Applications 80 (2022), 102023. The packet makes no inference beyond that cited scope.
Zhicheng Gao, Simon Kuttner, and Qiang Wang, “Counting irreducible polynomials with prescribed coefficients over a finite field”. Finite Fields and Their Applications 80 (2022), 102023. DOI 10.1016/j.ffa.2022.102023. Full journal article relevant to Most irreducible constant slices of a sparse quintic over F_101. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.For Most irreducible constant slices of a sparse quintic over F_101: Provides a source or neighboring result used in the packet for “irreducible quintic f101 slices.”
Packet source. Michael O. Rabin, Probabilistic Algorithms in Finite Fields, SIAM Journal on Computing 9 (1980), 273-280; exact executable sweep run on 2026-07-25. Complete exact sweep in iqf101-artifact-rabin-sweep. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Provides Rabin's finite-field irreducibility test used to cross-check the packet's complete computation over F_101.Also cited at Michael O. Rabin, Probabilistic Algorithms in Finite Fields, SIAM Journal on Computing 9 (1980), 273-280; exact executable sweep run on 2026-07-25.Source named by the research packet.

CC0 optimization over sparse polynomial slices.

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[#P2662] Most irreducible constant slices of a sparse quintic over F_101

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