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arxiv: 2605.20475 · v1 · pith:JW3KP6THnew · submitted 2026-05-19 · 🧮 math.NT

Bounded-box reductions in the Subbarao-Warren problem for unitary perfect numbers

Tom Maciejewski This is my paper

Pith reviewed 2026-05-21 06:29 UTC · model grok-4.3

classification 🧮 math.NT
keywords unitary perfect numbersSubbarao-Warren problemodd dependency graphsource kernels3-Higgs primescyclotomic polynomialsZsigmondy obstructions
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The pith

Three filters eliminate five impostor source kernels for unitary perfect numbers with seed exponents up to 10000.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper revisits the Subbarao-Warren problem for unitary perfect numbers by keeping the seed factor 2^a+1 explicit in the full balance equation. It shows that within a bounded enumeration of source components in the odd dependency graph, every admissible source kernel is either one of the two kernels from the known nonsquarefree examples or one of five additional impostor kernels. A reproducible three-filter certificate eliminates those impostors for all relevant seed classes with 1 <= a <= 10000. This reduces the search for new examples to checking an explicit finite frontier in the auxiliary set H_even of even m where every prime divisor of 2^m+1 is 3-Higgs.

Core claim

Within a bounded enumeration of source components in the odd dependency graph, every admissible source kernel is either one of the two kernels occurring in the known nonsquarefree examples, 3^2 and 5^4, or one of five additional impostor kernels. We give a reproducible three-filter certificate eliminating those impostor kernels for all relevant seed classes with 1 <= a <= 10000. The filters combine Zsigmondy-type exponent obstructions, inherited non-3-Higgs witnesses, and deterministic 2-adic budget overshoot. A structural lemma reduces finiteness of H_even to the prime branch m=2p, while computations prove |H_even ∩ [2,40000]| <= 201 and |H_even ∩ [2,50000]| <= 272 with explicit undecided 1

What carries the argument

the odd dependency graph of source components together with the three-filter certificate for eliminating impostor kernels

Load-bearing premise

The three filters together with the structural lemma capture all possible impostor kernels and correctly reduce the finiteness question without missing valid cases or introducing false negatives.

What would settle it

Discovery of a unitary perfect number with seed a between 1 and 10000 whose source kernel is one of the five impostors, or a direct counterexample to any of the three filters for some such a.

read the original abstract

A unitary perfect number is a positive integer n satisfying \sigma^*(n)=2n, where \sigma^* sums unitary divisors. Only five examples are known, and no sixth has been found. We revisit the Subbarao-Warren problem by keeping the seed factor 2^a+1 explicit in the full balance (2^a+1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}. Within a bounded enumeration of source components in the odd dependency graph, every admissible source kernel is either one of the two kernels occurring in the known nonsquarefree examples, 3^2 and 5^4, or one of five additional impostor kernels. We give a reproducible three-filter certificate eliminating those impostor kernels for all relevant seed classes with 1 <= a <= 10000. The filters combine Zsigmondy-type exponent obstructions, inherited non-3-Higgs witnesses, and deterministic 2-adic budget overshoot. The remaining obstruction is the auxiliary set H_even of even m for which every prime divisor of 2^m+1 is 3-Higgs. A structural lemma reduces finiteness of H_even to the prime branch m=2p, while allowing finite computations to leave composite candidates inherited from unresolved prime divisors. Using the supplied factor cache and APR-CL primality-verification transcripts, we prove |H_even \cap [2,40000]| <= 201 and |H_even \cap [2,50000]| <= 272, with explicit undecided frontier lists. Ford's theorem for downward-closed prime sets gives an unconditional power-saving thinness bound for H_even, but not finiteness. The remaining task is a divisor-level problem for the cyclotomic values \Phi_{4p}(2). Thus the paper does not prove finiteness; it gives a bounded-box elimination, a verified finite frontier, and a precise analytic target for closing the remaining branch.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to advance the Subbarao-Warren problem on unitary perfect numbers by performing a bounded enumeration of source components in the odd dependency graph for a up to 10000. It identifies that admissible source kernels are either the known ones from 3^2 and 5^4 or five impostor kernels, which are eliminated using a reproducible three-filter certificate consisting of Zsigmondy-type exponent obstructions, inherited non-3-Higgs witnesses, and deterministic 2-adic budget overshoot. Additionally, it bounds the set H_even of even m where every prime divisor of 2^m +1 is 3-Higgs, proving |H_even ∩ [2,40000]| ≤ 201 and |H_even ∩ [2,50000]| ≤ 272 with explicit lists, using a structural lemma to reduce finiteness to the prime branch m=2p, while providing a target for the divisor-level problem on Φ_{4p}(2).

Significance. If the three-filter certificate is complete and the computations are accurate, this work provides a substantial reduction in the possible forms for unitary perfect numbers, localizing the remaining obstruction to a specific analytic target. The use of reproducible certificates with APR-CL transcripts and the explicit frontier lists are notable strengths that facilitate further progress. The application of Ford's theorem for thinness bounds adds analytic context, though finiteness remains open.

major comments (2)
  1. § on the three-filter certificate: The completeness of the Zsigmondy-type exponent obstructions, inherited non-3-Higgs witnesses, and deterministic 2-adic budget overshoot (combined with the structural lemma) in eliminating all five impostor kernels for 1 ≤ a ≤ 10000 is load-bearing for the bounded-box reduction. The manuscript does not explicitly rule out the possibility that an admissible kernel could evade at least one filter via an unaccounted unitary divisor relation that undercounts the 2-adic overshoot, as noted in the stress-test concern; this requires a concrete verification or additional case analysis to confirm no false negatives occur.
  2. Section on H_even bounds and structural lemma: The reduction of finiteness of H_even to the prime branch m=2p allows composite candidates to be inherited from unresolved prime divisors. The explicit bounds |H_even ∩ [2,40000]| ≤ 201 and |H_even ∩ [2,50000]| ≤ 272 rely on the factor cache and APR-CL transcripts, but the manuscript should clarify how any such inherited composites are accounted for in the undecided frontier lists to ensure the counts are accurate.
minor comments (2)
  1. The notation for source kernels and the odd dependency graph would benefit from an explicit diagram or expanded definitions in the introductory sections to improve accessibility.
  2. Table or list of the five impostor kernels: include a brief summary of why each is admissible before filtering to aid readers in following the elimination process.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our bounded-box reduction for the Subbarao-Warren problem. The points raised help ensure the robustness of the three-filter certificate and the accuracy of the H_even frontier counts. We respond to each major comment below and have revised the manuscript accordingly to incorporate explicit verifications and clarifications.

read point-by-point responses

  1. Referee: § on the three-filter certificate: The completeness of the Zsigmondy-type exponent obstructions, inherited non-3-Higgs witnesses, and deterministic 2-adic budget overshoot (combined with the structural lemma) in eliminating all five impostor kernels for 1 ≤ a ≤ 10000 is load-bearing for the bounded-box reduction. The manuscript does not explicitly rule out the possibility that an admissible kernel could evade at least one filter via an unaccounted unitary divisor relation that undercounts the 2-adic overshoot, as noted in the stress-test concern; this requires a concrete verification or additional case analysis to confirm no false negatives occur.

    Authors: We agree that an explicit verification against potential evasion through unaccounted unitary divisor relations strengthens the load-bearing claim. The filters are designed so that Zsigmondy obstructions and non-3-Higgs witnesses operate independently of specific divisor configurations, while the 2-adic budget is always computed over the complete unitary divisor set of each kernel. To directly resolve the stress-test concern, the revised manuscript adds a dedicated case-analysis subsection that exhaustively checks each of the five impostor kernels for a ≤ 10000, confirming that no unitary divisor combination produces an undercount in the overshoot. This includes boundary simulations and reproduces the elimination for all seed classes. revision: yes

  2. Referee: Section on H_even bounds and structural lemma: The reduction of finiteness of H_even to the prime branch m=2p allows composite candidates to be inherited from unresolved prime divisors. The explicit bounds |H_even ∩ [2,40000]| ≤ 201 and |H_even ∩ [2,50000]| ≤ 272 rely on the factor cache and APR-CL transcripts, but the manuscript should clarify how any such inherited composites are accounted for in the undecided frontier lists to ensure the counts are accurate.

    Authors: The structural lemma explicitly states that composite candidates may be retained in the undecided lists when they derive from an unresolved prime p with 2p in the frontier. In the revised manuscript we have augmented the description of the frontier lists with a precise accounting rule: every composite entry is generated directly from the unresolved primes via the inheritance map, and we supply a supplementary data file that applies this rule to the explicit lists for both bounds. This ensures the reported cardinalities |H_even ∩ [2,40000]| ≤ 201 and |H_even ∩ [2,50000]| ≤ 272 correctly incorporate inherited composites without omission or double-counting, while remaining fully reproducible from the factor cache and APR-CL transcripts. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on external theorems and explicit computation

full rationale

The paper enumerates source components in the odd dependency graph up to explicit bounds (a ≤ 10000), applies three independent filters (Zsigmondy-type obstructions, non-3-Higgs witnesses, 2-adic budget) to eliminate five impostor kernels, and invokes Ford's theorem plus standard primality verification for the H_even bounds. The structural lemma reduces the finiteness question to the m=2p branch without defining the filters or the admissible kernels in terms of the final counts or results. No self-citations are load-bearing for uniqueness, no parameters are fitted to the target output, and no ansatz or renaming reduces the central claims to the inputs by construction. The work is self-contained against external benchmarks and presents explicit undecided frontier lists rather than claiming a closed proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard number-theoretic results (Zsigmondy theorem, Ford's theorem) and computational verification rather than introducing new free parameters, ad-hoc axioms, or invented entities.

axioms (1)
  • standard math Ford's theorem for downward-closed prime sets
    Invoked to obtain an unconditional power-saving thinness bound for H_even.

pith-pipeline@v0.9.0 · 5895 in / 1438 out tokens · 38665 ms · 2026-05-21T06:29:21.717013+00:00 · methodology

discussion (0)

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Reference graph

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