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

Carries and factorial digits

Instead of clearing the partial sum, we now track the only integer that could equal $m! S$ . Under rationality, $m! S$ is integral for all large $m$ ; since the positive tail is less than one, it must be the strict successor of $m! H_{m}$ . The recurrence for these successors therefore tests rationality without assuming that $m!$ clears $H_{m}$ . Put

Z_{m} = ⌊ m! H_{m} ⌋ + 1, Δ_{m} = Z_{m - 1} - (m - 1)! H_{m - 1} (m \geq 3) .

Thus $0 < Δ_{m} \leq 1$ , with $Δ_{m} = 1$ when the preceding scaled partial sum is integral. Define the integer carry $b_{m}$ by

\begin{matrix} (12) & Z_{m} = m Z_{m - 1} + 1 - b_{m} . \end{matrix}

For example, $2! H_{2} = 2$ and $3! H_{3} = 36 / 5$ , so $Z_{2} = 3$ and $Z_{3} = 8$ , giving $b_{3} = 3 \cdot 3 + 1 - 8 = 2$ . We use $⌊ x ⌋ + 1$ , not $⌈ x ⌉$ : at the integral value $2! H_{2} = 2$ , the required strict successor is $3$ .

Call $b_{m} = 1$ a unit carry. Put $E_{m} = m! (S - H_{m})$ and $ε_{m} = 1 / (m! - 1)$ . Since $d_{n + 1} > (n + 1) d_{n}$ , a geometric majorant gives the tail estimate

\begin{matrix} (13) & 0 < E_{m} < \frac{2 m!}{(m + 1)! - 1} < \frac{2}{m} \leq 1 (m \geq 2) . \end{matrix}

Compare the factorial scaling in Hančl and Tijdeman’s tail-integrality lemma for factorial series with integer coefficients [11]. Their scaled partial sums are integers; ours need not be, since already $3! H_{3} = 36 / 5$ . The next proof instead uses the short positive tail to identify the least integer above a scaled partial sum under a rationality assumption. The carry-defect expansion in §B.2 gives a separate, direct application of factorial-series rationality criteria.

Theorem 4.1 (an exact criterion from successive partial sums). For $m \geq 3$ ,

\begin{matrix} (14) & b_{m} = 1 ⟺ m ∣ Z_{m} ⟺ 1 + ε_{m} < m Δ_{m} \leq 2 + ε_{m} . \end{matrix}

Moreover

\begin{aligned} (15) & S \in Q & ⟺ b_{m} = 1 for all sufficiently large m, \\ (16) & S \notin Q & ⟺ \forall B \exists m > B : m ∤ Z_{m} . \end{aligned}

If $S = a / q$ with $a \in Z$ , $q \geq 1$ and $b_{m} \neq 1$ , then $q ∤ (m - 1)!$ and $q \geq m$ .

Proof. From $m! H_{m} = m Z_{m - 1} - m Δ_{m} + 1 + ε_{m}$ one gets $b_{m} = ⌈ m Δ_{m} - 1 - ε_{m} ⌉$ with $- 1 \leq b_{m} \leq m - 1$ , which gives both equivalences in (14), endpoints included.

Suppose $S = a / q$ and $q ∣ (m - 1)!$ . For $j = m - 1$ and $j = m$ the number $j! S$ is an integer, and $j! H_{j} = j! S - E_{j}$ with $0 < E_{j} < 1$ by (13), so $Z_{j} = j! S$ . Substituting into (12) forces $b_{m} = 1$ . Every fixed $q$ divides $(m - 1)!$ eventually, which gives one direction of (15) and, by contraposition, the final assertion; $q < m$ would imply $q ∣ (m - 1)!$ . Conversely, an eventual unit-carry tail makes $Z_{m} / m!$ eventually constant, and $0 < Z_{m} / m! - H_{m} \leq 1 / m!$ with $H_{m} \to S$ identifies that constant as $S$ , which is then rational. Negating (15) gives (16). ◻

The conclusion $q ∤ (m - 1)!$ is about prime-power multiplicities, not just prime factors. It permits all prime factors of $q$ to be at most $m - 1$ if one occurs to a larger exponent than in $(m - 1)!$ .

The adjacent-unit window condition

Factorial digits of 𝑆 −𝑒 +2S−e+2

Escape from a smaller interval

The series with denominators 𝑛! +𝑡n!+t

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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