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

The adjacent-unit window condition

Two successive unit carries imply a prime-power size condition after clearing denominators in their defining inequalities. The calculation below shows that the condition follows already from those carries. An additional arithmetic estimate is therefore needed to turn it into a contradiction.

Write $Δ_{k} = P_{k}^{Δ} / Q_{k}^{Δ}$ in lowest terms, with $P_{k}^{Δ}, Q_{k}^{Δ} > 0$ . In one recurrence step the unreduced denominator is $Q_{k}^{Δ} (k! - 1)$ ; let $G_{k} = Q_{k}^{Δ} (k! - 1) / Q_{k + 1}^{Δ}$ be the positive integer cancelled in reduction. For $m \geq 3$ , put

\begin{aligned} D_{m} & = Q_{m}^{Δ} (m! - 1) ((m + 1)! - 1), \\ Ω_{m} & = D_{m} (m (m + 1) Δ_{m} - m - 2 - \frac{m + 1}{m! - 1} - \frac{1}{(m + 1)! - 1}) . \end{aligned}

Both are integers. The two unit carries are equivalent to $0 < Ω_{m} \leq D_{m}$ . Independently of the carries, cancellation in two successive steps gives

D_{m} = Q_{m + 2}^{Δ} G_{m + 1} G_{m} = Q_{m}^{Δ} (m! - 1) ((m + 1)! - 1),

and under the pair assumption the offset factors to match, $Ω_{m} = P_{m + 2}^{Δ} G_{m + 1} G_{m}$ . Dividing by $G_{m + 1} G_{m} > 0$ reduces the proposed bound to

0 < P_{m + 2}^{Δ} \leq Q_{m + 2}^{Δ},

which every reduced positive gap in $(0, 1]$ already satisfies. Thus the resulting gap bound is not an additional restriction on a pair of unit carries. This calculation does not exclude an independent arithmetic obstruction involving the transition numerators or their valuations. Such an obstruction would need information not already implied by the pair assumption; it need not take the form of a bound on the preceding gap $Δ_{m}$ .

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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