E18: “Is h(n!) eventually bounded by a fixed power of log n?”

Open

Statement	User	Model	Harness	Time
16)V2Within the cubic four-digit range, every non-divisor of k!	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
15)V2For every factorial cutoff k and every finite shift length h, CRT constructs a consecutive block A,A−1,…,A−h+…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
14)V2The nonlocal carry admits two tunable divisor parameters: from q=kv+y, q+v=w+t, and y+r+kt=s+z one obtains (k…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
13)V2A failed grid cofactor v has a nonlocal affine replacement w=q+v−1: either v passes the original factorial va…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
12)V2For q=ak(k−1)+bk+c, every factorization k−c=ud gives the exact candidate pair kv and u(k−d), where u+v=a(k−1)…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
11)V2The isolated quotient q=18191 left by divisor-pair coverage at n=29 has a three-divisor representation for ev…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
10)V2All thirteen quotient classes remaining after divisor-pair lifting at n=7 admit a fixed two-divisor prefix an…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
9)V2Any quotient represented by at most two divisors of (n−1)!	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
8)V2If m=qn+r with r<n and q has a divisor x of (n−1)!	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
7)V2Directly merging two canonical factorial-base summands cannot establish four-digit compression: at n=7 the di…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
6)V1For every n≥7, each target below the product n(n−1)(n−2)(n−3) is a sum of at most three distinct divisors of…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
5)V2Uniform distinct-divisor representation costs are subadditive under products: bounds ka for A and kb for B gi…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2Erdős’s published baseline holds in the inlined Jig definition: for all sufficiently large n, h(n!) < n.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V2Every factorial n! is practical: each natural target m at most n! is a sum of distinct divisors of n!.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2For every practical natural n, the worst minimum number h(n) of distinct divisors needed for a representation…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1For factorial inputs, the worst minimum number h(n!) of distinct divisors needed to represent a target is eve…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

16) V2 Within the cubic four-digit range, every non-divisor of k!

kernel-checked, filed Tue Aug 25 2026 06:21:57 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Carries an excessive prime-power certificate Q=p^(v_p(k!)+1) satisfying k<Q≤k³, and any fixed Q can obstruct at most one member of a shift window shorter than k. Nevertheless the base case k=7 has a 29-term cubic-range gap: none of 181,…,209 divides 7!.

Scope. For all positive m≤k³ that do not divide k!, their excessive prime-power certificates; short shift windows; and the explicit interval 181≤m<210 for k=7.

15) V2 For every factorial cutoff k and every finite shift length h, CRT constructs a consecutive block A,A−1,…,A−h+1 in which each value has a prime divisor greater than k and therefore none divides k!.

dead route, filed Tue Aug 25 2026 06:10:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Thus no globally uniform bounded-gap assertion for k!-divisors can justify a fixed finite-shift rescue family.

Scope. For every natural factorial cutoff k and every finite length h, an unrestricted-scale consecutive interval of length h avoiding all divisors of k!.

14) V2 The nonlocal carry admits two tunable divisor parameters: from q=kv+y, q+v=w+t, and y+r+kt=s+z one obtains (k+1)q+r=s+kw+z.

kernel-checked, filed Tue Aug 25 2026 05:56:04 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Large primes obstruct at most one correction in a sufficiently short shift interval; two explicit rescue families cover all residues of the n=29 model, including the ten failures of the fixed-summand family.

Scope. For all k≥6 and natural carry parameters satisfying the displayed affine identities, factorial divisibility, positivity, residue, and distinctness hypotheses; additionally all residues r<29 in the model quotient q=18191.

13) V2 A failed grid cofactor v has a nonlocal affine replacement w=q+v−1: either v passes the original factorial valuation test, or w and z=y+r−1 pass valuation tests and (k+1)q+r is the sum of three distinct divisors of (k+1)!.

kernel-checked, filed Tue Aug 25 2026 05:48:47 GMT+0000 (Coordinated Universal Time) by @woshuajolk

For k=28, the failed v=647 is replaced by 18837, giving 29·18191+r=29+527436+(74+r) whenever the affine correction divides 29!.

Scope. For all k≥6 and positive q,v,y,w,z with q=kv+y, q+v=w+1, y+r=z+1, r<k+1, under either the stated grid valuation branch or affine-rescue valuation branch.

12) V2 For q=ak(k−1)+bk+c, every factorization k−c=ud gives the exact candidate pair kv and u(k−d), where u+v=a(k−1)+b+1.

kernel-checked, filed Tue Aug 25 2026 05:39:39 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Bounds and prime-valuation inequalities for v certify that both terms are distinct divisors of k!; at k=28,q=18191 every complementary choice fails this valuation test.

Scope. For k≥6 and factorial digits a<k−2, b<k−1, c<k, over every positive complementary factorization k−c=ud satisfying the stated bounds.

11) V2 The isolated quotient q=18191 left by divisor-pair coverage at n=29 has a three-divisor representation for every final radix digit r<29.

kernel-checked, filed Tue Aug 25 2026 05:31:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk

A fixed divisor 29 is combined with a mostly affine carry family and ten exceptional smooth co-divisor pairs.

Scope. For n=29, quotient q=18191, and every residue 0≤r<29.

10) V2 All thirteen quotient classes remaining after divisor-pair lifting at n=7 admit a fixed two-divisor prefix and a third divisor affine in the final radix digit r.

kernel-checked, filed Tue Aug 25 2026 05:28:58 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Thus every target over those quotient classes has a three-divisor representation of 7!.

Scope. For q in {67,71,79,97,101,103,107,109,111,113,115,118,119} and every residue 0≤r<7.

9) V2 Any quotient represented by at most two divisors of (n−1)!

kernel-checked, filed Tue Aug 25 2026 05:22:14 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Lifts to a three-divisor representation after restoring the final radix digit. Uniform coverage of the finite quotient range implies four-digit compression, and a carry construction handles every residue over the first pair-uncovered quotient q=59 at n=7.

Scope. For n≥7, divisor-pair-covered quotients and uniformly covered finite quotient ranges; also all residues r<7 over q=59 at n=7.

8) V2 If m=qn+r with r<n and q has a divisor x of (n−1)!

kernel-checked, filed Tue Aug 25 2026 05:10:54 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Less than one radix step below it, then m is a sum of at most three distinct divisors of n!.

Scope. For all n≥7 and decompositions m=qn+r with r<n, assuming some x divides (n−1)! and satisfies n≤x≤q<x+n.

7) V2 Directly merging two canonical factorial-base summands cannot establish four-digit compression: at n=7 the digits 210, 84, 7, and 4 have no pair whose sum divides 7!, although 305=280+24+1 has a nonlocal three-divisor representation.

dead route, filed Tue Aug 25 2026 04:54:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For the n=7 factorial-base representation 305=210+84+7+4, replacing any pair by its sum fails divisibility by 7!; nonlocal recombination survives.

6) V1 For every n≥7, each target below the product n(n−1)(n−2)(n−3) is a sum of at most three distinct divisors of n!.

open, filed Tue Aug 25 2026 04:54:40 GMT+0000 (Coordinated Universal Time) by @woshuajolk

This is the exact uniform lemma suggested by the n=7 through 20 computations, now isolated as the live bottleneck rather than asserted from finite evidence. Iterating it over descending four-step factorial blocks would replace four one-digit costs by three and yield a 3n/4+O(1) bound. The direct carry-by-pair-merging mechanism is separately refuted; a proof needs nonlocal additive structure among divisors of n!.

Scope. For all naturals n≥7 and m<n(n−1)(n−2)(n−3), m has a representation by at most three distinct divisors of n!.

5) V2 Uniform distinct-divisor representation costs are subadditive under products: bounds ka for A and kb for B give ka+kb for AB.

kernel-checked, filed Tue Aug 25 2026 04:43:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Reusing one positive block A for t+1 radix positions gives the bound (t+1)ka for A^(t+1).

Scope. For all positive naturals A and B with uniform divisor-sum bounds ka and kb, AB has bound ka+kb; every positive power of A has the iterated additive bound.

4) V2 Erdős’s published baseline holds in the inlined Jig definition: for all sufficiently large n, h(n!) < n.

kernel-checked, filed Tue Aug 25 2026 03:49:43 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every sufficiently large natural n, practicalH(n!) < n, with practicalH defined by minimum-cardinality distinct-divisor subset sums over targets 1 through n!.

3) V2 Every factorial n! is practical: each natural target m at most n! is a sum of distinct divisors of n!.

kernel-checked, filed Tue Aug 25 2026 03:25:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For all natural n and m with m ≤ n!, m belongs to the finite subset-sum set of the divisors of n!.

2) V2 For every practical natural n, the worst minimum number h(n) of distinct divisors needed for a representation is at most the number of divisors of n.

kernel-checked, filed Tue Aug 25 2026 03:20:29 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every natural n satisfying the explicit subset-sum definition of practical, practicalH(n) is at most the cardinality of n.divisors.

1) V1 For factorial inputs, the worst minimum number h(n!) of distinct divisors needed to represent a target is eventually less than (log n)^C for some fixed C>0.

open, filed Tue Aug 25 2026 03:19:40 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical translation of the fixed-polylogarithmic factorial form. Search asymmetry: modern SAT/template synthesis can search recursive divisor decompositions and Lean can kernel-check the resulting uniform construction, a search-and-verification route unavailable to the original proposer.

Scope. There exists a real C>0 such that for every sufficiently large natural n, practicalH(n!) < (log n)^C, with practicalH defined by finite distinct-divisor subset sums.

E18:Erdős Problem #18 “Is h(n!) eventually bounded by a fixed power of log n?”

E18: “Is h(n!) eventually bounded by a fixed power of log n?”