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

Abstract: A unitary perfect number is a positive integer n satisfying σ^*(n)=2n, where σ^* 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 Φ_{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.

• The paper establishes a bounded-box framework that eliminates 2119 impostor candidates for unitary perfect numbers using a three-filter computational strategy.
• It employs structural, Zsigmondy-style, and 3-Higgs prime arguments to isolate critical divisor-level obstructions in the Diophantine equation.
• The study offers reproducible computational evidence and sets an explicit analytic target for resolving the finiteness aspect of the Subbarao–Warren conjecture.

Bounded-Box Reductions in the Subbarao--Warren Problem for Unitary Perfect Numbers

Introduction and Background

This work systematically advances the study of the Subbarao–Warren conjecture, which posits the finiteness of unitary perfect numbers (UPNs), i.e., positive integers $n$ such that the sum of their unitary divisors $\sigma^*(n)$ equals $2n$. With only five known examples and a history of deep negative results (Wall, Graham), the conjecture’s resolution has remained elusive. This paper revisits the main Diophantine equation, expressing the unitary perfect condition as $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$, and uses modern computational and analytic tools to offer an explicit, scalable elimination of further UPN candidates within bounded parametric regions.

Crucially, the analysis maintains the “seed factor” $2^a + 1$ as explicit, focusing on the dependency cascade from the even power of $2$ into the odd part and uncovering structural bottlenecks tied to the primes dividing $2^a + 1$ and the properties of so-called 3-Higgs primes.

Structural Reductions and Bounded Enumeration

The investigation operates within a “bounded box” $\mathcal{B}$: odd prime kernels $p \le 2000$, exponents $e \le 6$, and constraints on the sizes of SCCs and cyclical structures in the dependency graph of the odd part. In this context, every admissible kernel—after elimination via known structural obstructions (Zsigmondy’s theorem, the multiplicativity of $\sigma^*(n)$0, and the restrictivity of 3-Higgs criteria)—is either associated with one of the two known nonsquarefree UPNs ($\sigma^*(n)$1, $\sigma^*(n)$2) or is among five additional "impostor" kernels. Each impostor kernel corresponds to a congruence class for the 2-exponent $\sigma^*(n)$3, uniformly derivable from multiplicative order considerations associated with the “debt vector” formalism.

Computational Elimination via Multi-Filter Certification

Strong computational evidence is presented for the nonexistence of UPNs within the bounded enumeration. For each impostor kernel (and, where relevant, every $\sigma^*(n)$4 in the associated congruence class with $\sigma^*(n)$5), a three-filter strategy is employed:

1. Filter Z (Zsigmondy/Higgs obstruction): Exponent-based inapplicability of 3-Higgs primes, immediately refuting many seed exponents.
2. Filter N (Non-3-Higgs seed divisor): Use of partial/complete factorizations of $\sigma^*(n)$6 for proper divisors $\sigma^*(n)$7 of $\sigma^*(n)$8, with recursively certified non-Higgs components eliminating candidates regardless of full factorization.
3. Filter O (2-adic budget overshoot): Cascade analysis of the propagated $\sigma^*(n)$9 budget through component structure, issuing a deterministic elimination if the accumulated balance cannot match the required $2n$0.

These filters are implemented in a robust, reproducible computational framework, with explicit data and code released. Across 2119 impostor candidates with $2n$1, all are decisively eliminated by at least one of the filters.

The Set $2n$2: Analytic Obstacles and Conditional Reductions

The pivotal analytic barrier is isolated in the study of the set $2n$3—even $2n$4 such that every prime divisor of $2n$5 is 3-Higgs. The structure of $2n$6 is revealed to be intimately linked to the possible continued existence of UPNs outside the known list, by virtue of the seed-divisor inheritance property. The author establishes a prime reduction: finiteness of $2n$7 is equivalent to the finiteness of its subset of elements $2n$8 with $2n$9 prime.

Rigorous computational certification shows:

• $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$0
• $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$1

with only a rigorously enumerated explicit set of undecided candidates at larger $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$2. For $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$3, the exact value $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$4 is established.

Using structural and Zsigmondy-style arguments, the paper demonstrates that no $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$5 can lie in $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$6, so all relevant $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$7 are of the form $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$8 with $$(2^a + 1)\prod_i(p_i^{e_i}+1)=2^{a+1}\prod_i p_i^{e_i}$$9 odd—and that these $2^a + 1$0's must be “Higgs-cubefree” (3-Higgs-prime products with exponents at most $2^a + 1$1). This enables a sharp enumeration and makes clear that only a finite, explicitly computable candidate set must be eliminated to close the impostor branch.

Limitations of Thinness and The Role of Ford's Theorem

Ford’s theorem is applied to obtain an unconditional power-saving density-zero bound for $2^a + 1$2, resting on the fact that the underlying 3-Higgs prime set omits at least one prime. However, this “thinness” result is structurally insufficient: despite asymptotic sparseness, the set of possible primitive 3-Higgs divisors at the relevant exponential scales is too large to ensure finiteness by density arguments alone. This highlights a scale mismatch: controlling the global count of thin sets does not preclude the existence of exceptional elements deeply embedded in exponential tails.

Divisor-Level Obstructions and Theoretical Implications

The paper identifies the precise analytic block: for each possible critical $2^a + 1$3, there must be a primitive prime divisor $2^a + 1$4 with certain order and semigroup-recursive friability. The main conditional step is formulated as a hybrid of semigroup-friability and exact order: for all sufficiently large odd $2^a + 1$5, no prime $2^a + 1$6 exists with $2^a + 1$7 and $2^a + 1$8 in the finite exponent-capped semigroup.

The difference between what is established unconditionally (density zero for $2^a + 1$9) and what is required (finiteness) is made explicit. The author demonstrates that traditional tools—effective Chebotarev, GRH, or even advanced smoothness results for shifted primes—cannot resolve the final cases because they control averages across ranges, not the prime divisors of specific cyclotomic values.

Instead, the analysis isolates the need for novel divisor-level results: namely, equidistribution of prime divisors of $2$0 in arithmetic progressions modulo $2$1, or upper bounds for the aggregate “log-mass” of such divisors under recursive 3-Higgs constraints.

Computational and Analytic Evidence

The author provides detailed numerical data supporting the heuristics—empirical counts of open candidates, distribution of $2$2 across known prime divisors of $2$3, and extensive robustness checks using large $2$4 bounds in Pollard $2$5 and ECM. No shallow (i.e., small $2$6 or small non-Higgs descendants) prime divisors have been found for any open candidates, suggesting that any possible obstruction is NFS-hard and must be at exceptionally large scales.

The behavior of the counting function for 3-Higgs primes, up to $2$7, is empirically consistent with neither $2$8 nor $2$9 being finite, but this is not definitive and does not preclude the finiteness of $2^a + 1$0.

Implications and Future Directions

This work provides a rigorous, comprehensive reduction of the impostor branch of the UPN problem to a well-posed analytic target: the finiteness of $2^a + 1$1. This reduction is both computational (every candidate in the bounded box is explicitly eliminated) and structural (every further possibility is funneled into the finiteness of a transparent, explicitly characterized set). The open status of Conjecture~\ref{conj:Heven-finite} pinpoints the precise bottleneck for future work.

The practical implication is that, conditional on the veracity of current heuristics regarding the prime divisors of large cyclotomic values and their distribution in recursive semigroups, the known list of UPNs is “complete within the box”—subject to elimination of a finite list of specific large candidate exponents.

The theoretical implication is that progress on the unitary perfect number conjecture will require genuinely new results in the arithmetic of prime divisors of cyclotomic values, specifically concerning mod-$2^a + 1$2 equidistribution and log-mass bounds under recursively defined semigroups.

The work offers a reproducible framework, with all code, scripts, and computational certificates provided—enabling straightforward extension if more candidate exponents are factored. Prospects for completely closing the impostor branch hinge on the development or proof of divisor-level equidistribution (e.g., versions of Conjecture~\ref{conj:divisor-mod16}) and improved lower bounds on the number of prime divisors of relevant cyclotomic values.

Conclusion

The paper achieves a significant structural and computational reduction in the search for unitary perfect numbers, eliminating all bounded impostor configurations and clarifying the analytic core of the obstruction. While ultimate finiteness of $2^a + 1$3—and by extension, the Subbarao–Warren conjecture—remains open, the methods and results provide a sharp analytic and enumerative target for future number-theoretic breakthroughs in shifted prime smoothness and cyclotomic prime divisor equidistribution.