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

Finite vectors and exact numerical examples

The three depth-four vectors below all have $M = 1380$ . By Theorem 1.3, their remainders differ only by integers. A single finite calculation therefore tests all three:

255 + \frac{4}{5} < 242 + 1380 \sum_{d = 5}^{8} \frac{1}{d! - 1} < 255 + \frac{5}{6}, \frac{2 \cdot 1380}{9! - 1} < \frac{1}{100} .

The first remainder lies strictly between $255$ and $256$ . Subtracting the corresponding integer offsets gives the other two intervals.

Vector	$\sum_{n > 4} z_{n}$	$k$	Interval for $R$
$12 K_{4} + 253 U_{6} - 11 U_{8}$	$242$	$- 1474$	$(255, 256)$
The five-term vector (5)	$- 44$	$- 1760$	$(- 31, - 30)$
The norm minimizer on ${2, \dots, 8}$	$- 56$	$- 1772$	$(- 43, - 42)$

The second column is the offset from $1380 (S - H_{4})$ ; the third is the integer $k$ in $R = 1380 S + k$ . Since $1380 H_{4} = 1716$ , subtracting $1716$ converts the second column into the third. The five-term remainder is $286$ less than the first, and the norm-minimising remainder is another $12$ less. All three exclude exactly the same divisors of $1380$ by this argument. The coefficient and support improvements in §1.2 do not enlarge that set of exclusions; the larger finite computations in §6 already imply it. The exact offsets are reproduced by scripts/check_erdos68_channel_moment.py.

The finite-support vector $λ = 2 e_{3} - e_{4}$ has, by kernel check, $V_{2} (λ) = 0$ , $M = - 12$ , $V_{3} (λ) = - 2$ , $V_{4} (λ) = 11$ , and $V_{d} (λ) = - 12$ for every $d \geq 5$ . Under the exact tail enclosure $1 / 119 < \sum_{d \geq 5} 1 / (d! - 1) < 1 / 50$ its remainder lies strictly between $- 93 / 575$ and $- 309 / 13685$ , so it is nonzero and has absolute value less than $1$ .

In the enlarged coefficient space of §1, exact integer computation verifies the vectors $K_{D}$ for every $2 \leq D \leq 12$ : weighted sums $2$ through $D$ vanish, $M = L_{D}$ , and the coefficients have gcd one. They lie outside the space of vectors supported on $n \geq 2$ , since $L_{D}$ is odd whereas $M$ is even for every such vector. At $D = 9$ ,

L_{9} = 31540008254514077395, a_{9} = [e_{1}] K_{9} = - 3902884074990939115 .

Since $U_{11} = T_{11}$ has $u_{11} = 0$ , the vector $K_{9} - 9553024718754 U_{11}$ keeps the coordinate $a_{9}$ and has coefficients with gcd one. By Theorem 1.3 its residual is $L_{9} (S - H_{9}) - 9553024718754$ , and exact rational arithmetic with the tail bound $\sum_{n \geq 36} 1 / (n! - 1) < 2 / (36! - 1)$ places it strictly between $1353 / 100000$ and $1354 / 100000$ . A different example is supported on $n \geq 2$ : the vector $c = (- 40, 55, - 10, 1)$ on the support $(3, 4, 5, 6)$ annihilates weighted sums $2$ and $3$ , has $M = 600$ , and satisfies

0.09925341997208298 < R (c) < 0.09925341997208300,

which excludes denominators dividing $600$ .

A separate interval computation reports the stronger geometric statement that no lower-interval event $m θ_{m - 1} < E_{m}$ occurs at any $3 \leq m \leq 100000$ ; its executable and source digest are not available, so that classification remains external finite evidence. The carry computation through $300000$ used in the short note is the exact GMP computation recorded in Section 6, with the separate calculation through $4000$ ; it is computational evidence outside the Lean development.

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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