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

TL;DR

This paper investigates the Subbarao-Warren problem related to unitary perfect numbers, providing a bounded-box elimination method, verified frontiers, and an analytic target, but does not prove finiteness.

Contribution

It introduces a reproducible filtering approach to eliminate impostor kernels and bounds the set H_even, advancing the understanding of the problem's finite aspects.

Findings

Eliminated impostor kernels for seed classes with 1 <= a <= 10000.

Bounded the size of H_even within specific ranges, e.g., <= 201 for [2,40000].

Provided a verified finite frontier and an analytic target for future work.

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.