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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Factorial digits of $S - e + 2$

The small perturbation from the reciprocal-factorial series can also be isolated term by term. Subtracting $e - 2$ leaves the absolutely convergent correction $C = \sum_{n \geq 2} (n! (n! - 1))^{- 1}$ . The identity $1 / (n! - 1) = 1 / n! + 1 / (n! (n! - 1))$ gives

\begin{matrix} (17) & S = C + e - 2 . \end{matrix}

The canonical factorial digits of a real $x$ are $a_{m} (x) = ⌊ m! x ⌋ - m ⌊ (m - 1)! x ⌋ \in {0, \dots, m - 1}$ for $m \geq 2$ . At a rational endpoint, the floor convention selects the terminating representation. For example, $1 / 2 = 1 / 2!$ also equals $\sum_{m \geq 3} (m - 1) / m!$ , but its canonical digits are $a_{2} = 1$ and $a_{m} = 0$ for $m \geq 3$ , not an eventually maximal tail. With this convention, $x$ is rational exactly when its digits vanish from some index on. The criterion goes back to Cantor [5]; Galambos treats the rationality of Cantor series in [6], and Koepf and Schmersau prove the irrationality direction for nonterminating factorial expansions whose digits are not eventually maximal [9].

Proposition 4.2 (rationality and factorial residues).

S \in Q ⟺ ⌊ m! C ⌋ \equiv - 2 (\mod m) for all sufficiently large m .

Proof. Put $J_{m} = \sum_{k = 0}^{m} m! / k!$ . Then $J_{m}$ is an integer, every summand with $k < m$ is divisible by $m$ and the summand at $k = m$ is $1$ , so $J_{m} \equiv 1 (\mod m)$ ; and $0 < m! e - J_{m} < 1$ for $m \geq 2$ .

Suppose $S$ is rational. For all large $m$ both $(m - 1)! S$ and $m! S$ are integers, so $m ∣ m! S$ . Multiplying (17) by $m!$ and using $⌊ - m! e ⌋ = - J_{m} - 1$ gives $⌊ m! C ⌋ = m! S + 2 m! - J_{m} - 1 \equiv - 2 (\mod m)$ .

Conversely, assume the congruence from some index on. Because $0 \leq a_{m} (C) < m$ , for $m \geq 3$ the congruence is equivalent to $a_{m} (C) = m - 2$ . Choose $N$ beyond the exceptional indices. The canonical expansion gives

C = \frac{⌊ N! C ⌋}{N!} + \sum_{m > N} \frac{m - 2}{m!},

and the telescope $\sum_{m > N} (m - 1) / m! = 1 / N!$ turns this into

S = C + e - 2 = \frac{⌊ N! C ⌋ + 1}{N!} + \sum_{m = 2}^{N} \frac{1}{m!},

which is rational. ◻

The proof applies to every real $C$ , not just this series: eventual digits $a_{m} (C) = m - 2$ characterize rationality of $C + e - 2$ . Thus $C = 2 - e$ satisfies the eventual pattern, whereas $C = 0$ does not. For our particular positive $C$ , neither possibility is decided by a finite digit list.

Negating the eventual congruence gives the equivalent formulation

\begin{matrix} (18) & S \notin Q ⟺ (\forall B) (\exists m > B) (⌊ m! C ⌋ + 2) mod m \neq 0. \end{matrix}

It remains to prove failures of this congruence at arbitrarily large indices for this particular $C$ . A finite list of failures does not settle this question, and the equivalence gives no distribution estimate.

Recovering the finite carries.

To compare canonical digits with the carries, we must account for the tail removed before taking a floor. Set

δ_{m} = m! \sum_{n > m} \frac{1}{n! (n! - 1)}, σ_{m} = 1_{{{m! C} < δ_{m}}} .

Here ${x} = x - ⌊ x ⌋$ . The factorial-tail bound gives $0 < δ_{m} < 1 / ((m + 1)! - 1) < 1$ . Subtracting $δ_{m}$ from $m! C$ lowers its floor by one exactly when ${m! C} < δ_{m}$ . Therefore

Z_{m} = m! \sum_{n = 2}^{m} \frac{1}{n!} + ⌊ m! C ⌋ + 1 - σ_{m} .

Substitution in (12) gives

\begin{matrix} (19) & b_{m} = m - 1 - a_{m} (C) + σ_{m} - m σ_{m - 1} (m \geq 3) . \end{matrix}

When ${m! C} = δ_{m}$ , the scaled partial sum is integral and $σ_{m} = 0$ . For a smaller fractional part, subtracting even this small tail crosses an integer. Tail size alone therefore permits neither correction term to be omitted.

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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Factorial digits of 𝑆 −𝑒 +2S−e+2

Recovering the finite carries.

About this paper

Factorial digits of $S - e + 2$