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

Integer coefficient vectors and cancellation

We need weights that preserve the fractional part of $i! / (d! - 1)$ and whose scaled adjacent differences have a simple form. Both requirements are met by removing one factor $d!$ for each complete block of $d$ indices. For a finitely supported integer vector $c = (c_{i})$ on $i \geq 2$ , put

M (c) = \sum_{i} c_{i} i!, W_{d, i} = \frac{i!}{(d!)^{⌊ i / d ⌋}}, V_{d} (c) = \sum_{i} c_{i} W_{d, i} (d \geq 2) .

We allow either sign for $M (c)$ and for $V_{d} (c)$ . The associated remainder is

R (c) = \sum_{d \geq 2} \frac{V_{d} (c)}{d! - 1} .

The index $i$ selects a coefficient, whereas $d$ selects a summand of the remainder. If $c$ is supported on $i \leq N$ , then for $d > N$ all floor exponents vanish and $V_{d} (c) = M (c)$ . The series therefore converges absolutely. Cancellation of the first weighted sums leaves a finite signed sum followed by $M (c)$ times the original tail. We show below that the remainder differs from $M (c) S$ by an integer.

The chosen exponent is a floor, rather than the largest possible power of $d!$ dividing $i!$ . For example, at $d = 2$ , $i = 6$ , it is $3$ although $2^{4} ∣ 6!$ . What matters is that it changes between $i - 1$ and $i$ exactly when $d ∣ i$ . This makes adjacent differences amenable to divisor elimination in §1.1. Each $W_{d, i}$ is an integer: writing $i = k d + r$ with $0 \leq r < d$ , the quotient $i! / ((d!)^{k} r!)$ is a multinomial coefficient. Since $i! = (d!)^{⌊ i / d ⌋} W_{d, i}$ and $d! \equiv 1 (\mod d! - 1)$ ,

\begin{matrix} (1) & V_{d} (c) \equiv M (c) (\mod d! - 1) . \end{matrix}

Theorem 1.1 (divisibility of the difference). For every finite integer support and every $d \geq 2$ there is an integer $k$ with $V_{d} (c) = M (c) + (d! - 1) k$ .

Proof. By (1), $d! - 1$ divides $V_{d} (c) - M (c)$ . Thus $k = (V_{d} (c) - M (c)) / (d! - 1)$ is an integer and gives the identity. The modulus is positive also at $d = 2$ , when it equals one. ◻

For any $N \geq 2$ at least as large as every supported index, the terms with $d > N$ equal $M (c) / (d! - 1)$ . Therefore

R (c) - M (c) S = \sum_{d = 2}^{N} \frac{V_{d} (c) - M (c)}{d! - 1} \in Z .

The sum is finite and each term is integral by (1). This argument also holds when index $1$ is temporarily allowed in the basis calculation below. In particular, $M = 0$ gives an integral remainder, and fixing $M (c)$ fixes its fractional part. These conclusions hold for every finite vector, before imposing cancellation.

By (1), a vanishing $d$ -th weighted sum forces $(d! - 1) ∣ M (c)$ , and annihilating every weighted sum $2 \leq d \leq D$ forces $L_{D} ∣ M (c)$ , where $L_{D} = {lcm}_{2 \leq d \leq D} (d! - 1)$ is the same quantity as in §3. With $M$ fixed, the same congruence fixes the fractional part of each quotient $V_{d} (c) / (d! - 1)$ separately.

An integral basis and the possible values of 𝑀M

Examples and support refinements

Support bounds and nonintegrality

A primitive solution on an arithmetic progression

Cofactors give the same primitive vector

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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