Bounded-box reductions in the Subbarao-Warren problem for unitary perfect numbers
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| Format: | Preprint |
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2026
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| _version_ | 1866918519856168960 |
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| author | Maciejewski, Tom |
| author_facet | Maciejewski, Tom |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_20475 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bounded-box reductions in the Subbarao-Warren problem for unitary perfect numbers Maciejewski, Tom Number Theory 11A25, 11Y05, 11N69, 11Y50, 11A41 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. |
| title | Bounded-box reductions in the Subbarao-Warren problem for unitary perfect numbers |
| topic | Number Theory 11A25, 11Y05, 11N69, 11Y50, 11A41 |
| url | https://arxiv.org/abs/2605.20475 |