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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The series with denominators $n! + t$

The digit argument also applies to the other shifts mentioned by Erdős. For an integer $t \geq - 1$ , put $S_{t} = \sum_{n \geq 2} 1 / (n! + t)$ and $C_{t} = \sum_{n \geq 2} 1 / (n! (n! + t))$ , so that $S_{t} = - t C_{t} + (e - 2)$ .

The restriction $t \geq - 1$ keeps every denominator positive for $n \geq 2$ ; $t = - 2$ is excluded because its first denominator is zero. All series in this identity converge absolutely. The familiar case $t = 0$ is $e - 2$ ; this example will make the residue condition explicit.

Theorem 4.4 (a criterion for the shifts $t \geq - 1$ ). For every integer $t \geq - 1$ , the series $S_{t}$ is rational exactly when

⌈ t m! C_{t} ⌉ \equiv 2 (\mod m)

for all sufficiently large $m$ , and irrational exactly when that residue is missed cofinally. The member $t = - 1$ is $S$ , and the member $t = 0$ is $e - 2$ , for which the scaled correction is always $0$ and hence misses the residue class at every $m \geq 3$ , proving the irrationality of $e$ .

Proof. Put $Y = - t C_{t}$ , so that $S_{t} = Y + e - 2$ . The proof of Proposition 4.2 applies to any real $Y$ : rationality of $Y + e - 2$ forces $⌊ m! Y ⌋ \equiv - 2 (\mod m)$ eventually. Conversely, that congruence forces the canonical digits of $Y$ to equal $m - 2$ eventually; adding the factorial series of $e - 2$ leaves an eventually telescoping tail, hence a rational sum. Finally $⌈ t m! C_{t} ⌉ = - ⌊ m! Y ⌋$ translates the congruence. Negation gives the cofinal statement. For $t = 0$ , the left side is zero, which is not congruent to $2$ for $m \geq 3$ . ◻

The $t = 0$ specialisation recovers the classical irrationality of $e$ . For $t \neq 0$ , no estimate proving failure of the displayed congruence at arbitrarily large indices is supplied here. This is a statement about the scope of the present argument, not an assertion that every other member has the same current literature status.

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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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The series with denominators 𝑛! +𝑡n!+t

About this paper

The series with denominators $n! + t$