Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria

Will Cook

Long paper

Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria

Erdős #68 By Will Cook 1 September 2026; revised 7 October 2026 Authorship, AI use and citation

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Summary. The record proves liminf (log L_N)/(N^3/2 log N) $\geq 2 \sqrt{2} / 3$ for L_N=lcm(2!−1,…,N!−1), by comparing a terminal block of denominators with pairwise gcds rather than an external multiplicity theorem. Specialising a Louwsma–Martino valuation, it exhibits primes 139 and 2593 absent from all later reduced denominators. Coefficient constructions give integer forms MS+k; at fixed moment the remainder changes only by an integer. Rationality criteria use factorial digits and the next integer above a scaled partial sum; the digit argument also recovers the irrationality of e. Two exact calculations give q∤299999! and $q \geq 2^{39990} > 10^{12038}$ . These exclusions do not prove irrationality.

This paper covers the complete problem-specific reasoning surface for Erdős #68, including all registered result families and their boundaries.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #68, which remains open.

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In this paper

Limits of the recurrences and clearing factors used here

Two distinctions are needed in using these identities: a recurrence may have solutions that violate the defining floor relations, and a fixed factor may eventually be absorbed by a growing factorial. The examples below keep these cases separate.

Without the defining floor relation, the carry recurrence and its range allow $Z_{m} / m!$ to be the constant $3 / 2$ . Taking $Z_{m} = 3 m! / 2$ for $m \geq 2$ and $b_{m} = 1$ for $m \geq 3$ gives $Z_{m} = m Z_{m - 1} + 1 - b_{m}$ and $- 1 \leq b_{m} \leq m - 1$ , with the actual initial value $Z_{2} = 3$ . But it gives $Z_{3} = 9$ , whereas the actual partial sum $H_{3} = 6 / 5$ gives $Z_{3} = 8$ . This example only disproves sufficiency of the stated recurrence, bounds and initial integer. It fails the floor definition. No claim is made that it satisfies the additional prime-index identities. The rational recurrence for $F_{m} = m! H_{m}$ in §B.1 determines the actual partial sums uniquely.
The arguments considered with Wilson quotients, harmonic sums, $p$ -adic gamma identities, and factorial residues still need a real gap estimate and the required modular divisibility at the same indices. No such unbounded family is obtained here. This records the missing step in these arguments, not an impossibility theorem for the use of those identities.
For the genus-zero product $E (z) = \prod_{n \geq 2} (1 - z / n!)$ , local uniform convergence and logarithmic differentiation give $- E^{'} (1) / E (1) = S$ . Termwise clearing by $\prod_{n = 2}^{N} (n! - 1)$ cannot give a positive remainder tending to zero: this product is at least $L_{N}$ , and $L_{N} (S - H_{N}) \to \infty$ by §3. The conclusion concerns this clearing factor alone; Hermite–Padé systems with other denominators are not ruled out.
Changing a coefficient vector without changing its value of $M$ changes the remainder by an integer only. Also, cancelling the first $D - 1$ weighted sums forces $L_{D}$ to divide the value of $M$ ; factorial divisibility does not remove that constraint.
A fixed pair of denominator indices cannot make the projection argument work at arbitrarily large parameters, because their factors eventually divide the factorial being removed.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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