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

Relations among the criteria

The classification determines all cancellation vectors and attainable multipliers. The progression construction adds eventual divisibility by each fixed $q$ and a vanishing tail bound for the same vector. Its primitive cofactor construction gives that vector up to sign, not another family. At fixed $M$ , coefficient changes still alter only the integer in $R (c) = M (c) S + k$ .

Common-denominator growth measures the cost of clearing, while prime-power survival describes reduction. Neither supplies the missing gap comparison. The carry, digit and interval criteria characterize irrationality but leave their required unbounded departures unproved. The finite exclusions restrict divisibility and size, respectively; they establish no unbounded assertion.

The remaining target for this method is therefore precise: for every $q > 0$ , find a nonintegral remainder whose own multiplier satisfies $q ∣ M$ . The algebraic classification supplies choices, not this arithmetic conclusion.

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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