Maciejewski 2026 bounded box reductions subbarao warren problem

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Tom Maciejewski, Bounded-box reductions in the Subbarao-Warren problem for unitary perfect numbers. arXiv preprint (2026). arXiv:2605.20475. The copy read for this card is arXiv v2 (24 May 2026). The arXiv record (https://arxiv.org/abs/2605.20475, read 2026-10-07) names the Creative Commons Attribution 4.0 license.

The paper attacks the conjecture that 6, 60, 90, 87360 and 146361946186458562560000 are the only unitary perfect numbers, working from the full balance (2^a+1) prod (p_i^{e_i}+1) = 2^{a+1} prod p_i^{e_i} with the seed factor 2^a+1 kept explicit. The kernels are the small source configurations of the odd dependency graph, with kernel primes at most 2000 in the enumeration box. Within that box, the only admissible source kernels are the two kernels of the known nonsquarefree examples, 3^2 and 5^4, and five further impostor kernels. For each a with 1 <= a <= 10000 in an impostor kernel's seed class, at least one of three filters excludes it (Theorem 3 and Corollary 4): exponent obstructions of Zsigmondy type, a prime divisor of 2^m+1 that is not 3-Higgs, or an excess in the 2-adic balance. What remains is the set H_even of even m for which every prime divisor of 2^m+1 is 3-Higgs. By Proposition 5, if m = 2k is in H_even with k odd, then k is a product of 3-Higgs primes each to power at most 3, and 2d is in H_even for every odd divisor d of k; Theorem 8 deduces that H_even is finite if and only if its prime branch, the m = 2p in H_even, is finite. The verified computational bounds are |H_even cap [2,40000]| <= 201 and |H_even cap [2,50000]| <= 272. Let H be the set of all m >= 1 for which every prime divisor of 2^m+1 is 3-Higgs, so that H_even is its even part. Theorem 22 bounds the number of m <= X in H, and so in H_even, by O(X^{1-eta}) for an absolute eta > 0, using Ford's theorem on downward-closed prime sets; it does not give finiteness. Theorems 28 and 31 are conditional on analytic hypotheses. The paper does not prove the full conjecture; in the abstract's words (p. 1), "it supplies a bounded-box elimination, a finite verified frontier, and a precise analytic target for the remaining obstruction." This is the May 2026 substantial-progress item recorded for problem 1052.

Source: https://arxiv.org/abs/2605.20475.

Bears on. #1052

Results to transcribe.

• Proposition 1: Every prime divisor of a unitary perfect number is 3-Higgs. The paper's list of its rigorous results (§1) does not include it, and its printed proof uses q | p-1 | p^e+1 for odd e, which fails in general (p = 5, e = 1 gives 4 and 6).
• Proposition 5: Structural lemma: if m = 2k lies in H_even with k odd, then every prime factor of k is 3-Higgs and divides k at most to the third power, and 2d lies in H_even for every odd divisor d of k.
• Theorem 8: H_even is finite if and only if its prime branch {m = 2p in H_even : p odd prime} is finite; if that branch has N elements, then |H_even| <= 4^N.
• Theorem 22: For an absolute eta > 0, the number of m <= X in H (the m for which every prime divisor of 2^m+1 is 3-Higgs) is O(X^{1-eta}), and the sum of 1/m over H converges; the same holds for H_even and H_odd. The proof uses Ford's theorem for downward-closed prime sets.
• Computational bound: |H_even cap [2,40000]| <= 201 and |H_even cap [2,50000]| <= 272, with explicit undecided candidate lists and APR-CL verified witness primes.
• Theorem 28 / Theorem 31: Conjectures 26 and 27 together imply that H_even is finite (Theorem 28); H_even is finite under two effective hypotheses on the prime divisors of Phi_{4p}(2), one of Chebotarev type and one on the growth of their number (Theorem 31).