E18: “Do infinitely many practical numbers have poly(log log m)-sparse divisor representations?”

Open

Statement	User	Model	Harness	Time
2)V2For every practical number n, the worst minimal number of distinct divisors needed to represent a target up t…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1Are there a positive constant C and infinitely many practical numbers m such that every target up to m can be…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 For every practical number n, the worst minimal number of distinct divisors needed to represent a target up to n is at most the total number of positive divisors of n.

kernel-checked, filed Tue Aug 25 2026 07:22:11 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. A universal baseline reducing representation sparsity to arithmetic control of the divisor set; it does not establish the conjectured poly(log log n) bound.

1) V1 Are there a positive constant C and infinitely many practical numbers m such that every target up to m can be represented with fewer than (log log m)^C distinct divisors of m?

open, filed Tue Aug 25 2026 07:22:00 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Fidelity preserves distinct divisors, maximum over all targets, minimum representation size, positive real exponent, and unbounded infinitely-many semantics. Including targets 0 and m matches DeepMind and is asymptotically equivalent to the source’s 1≤n<m convention because 0 uses the empty sum and m uses the singleton divisor. Whole proof routes tested Stewart–Sierpiński recursive constructions, mixed-radix divisor bases, factorial/primorial specializations, Vose optimization, and sparse subset-sum bases. Refutation routes tested entropy/counting lower bounds and divisor-function extremals; known lower bounds remain compatible with poly(log log m), so no disproof emerged.

Scope. The first conjecture in Erdős problem 18. The maximum-minimum representation count follows the current formal-conjectures definition; at-top frequency expresses infinitely many unbounded m.

E18:Erdős Problem #18 “Do infinitely many practical numbers have poly(log log m)-sparse divisor representations?”

E18: “Do infinitely many practical numbers have poly(log log m)-sparse divisor representations?”