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.

Start here, by the editorial route

In this paper

Growth of the common denominator

A large product of denominators gives a large least common multiple only if their overlap is controlled. We use the last $k$ denominators: their logarithmic product contributes about $k N \log N$ , while the pairwise-gcd estimate costs about $(k^{3} / 6) \log N$ . With $k \sim α \sqrt{N}$ , the leading coefficient is $α - α^{3} / 6$ , maximized at $α = \sqrt{2}$ . The lemmas below make this balance rigorous and yield $lim inf \log L_{N} / (N^{3 / 2} \log N) \geq 2 \sqrt{2} / 3$ . Here $L_{N}$ is the common denominator before reduction, not $den (H_{N})$ .

Lemma 3.1 (product, least common multiple, pairwise gcd). For positive integers $x_{1}, \dots, x_{k}$ ,

\prod_{i = 1}^{k} x_{i} | lcm (x_{1}, \dots, x_{k}) \prod_{i < j} gcd (x_{i}, x_{j}) .

Proof. For each prime $r$ , arrange $a_{i} = v_{r} (x_{i})$ in increasing order. The valuation of the right-hand side is $a_{k} + \sum_{i = 1}^{k - 1} (k - i) a_{i}$ , whereas that of the left-hand side is $\sum_{i = 1}^{k} a_{i}$ . Their difference is $\sum_{i = 1}^{k - 1} (k - i - 1) a_{i} \geq 0$ . This proves divisibility at every prime. ◻

The next divisibility is the case $P = - 1$ of the relation $gcd (i! + P (i), j! + P (j)) ∣ (j! / i!) P (i) - P (j)$ for $P \in Z [X]$ , obtained by multiplying $i! + P (i)$ by $j! / i!$ and subtracting $j! + P (j)$ . Luca and Shparlinski use this relation to bound a common divisor of $n! + P (n)$ and $(n + h)! + P (n + h)$ in the proof of their Lemma 5 [13], and Lai uses it for a common prime-power divisor [14].

The earlier spacing method of Erdős and Stewart [18] is related background. We now state only the subtraction needed for the lcm argument, rather than importing the prime-factor estimates of those papers.

Lemma 3.2 (factorial-gap gcd). For $2 \leq i < j$ , the integer $g = gcd (i! - 1, j! - 1)$ divides $j! / i! - 1$ , and $g \leq j! / i! - 1 < j^{j - i}$ .

Proof. The identity $(j! / i!) (i! - 1) - (j! - 1) = 1 - j! / i!$ shows that $g$ divides $j! / i! - 1$ . This integer is positive. Moreover, $j! / i!$ is a product of $j - i$ integers at most $j$ , so $g \leq j! / i! - 1 < j^{j - i}$ . ◻

Lemma 3.3 (segment inequality). For $2 \leq k \leq N - 1$ ,

\begin{matrix} (10) & \sum_{n = N - k + 1}^{N} \log (n! - 1) \leq \log L_{N} + (\binom{k + 1}{3}) \log N . \end{matrix}

Proof. Apply Lemma 3.1 to the $k$ denominators with indices $N - k + 1, \dots, N$ . Their lcm divides $L_{N}$ . Lemma 3.2 bounds the logarithm of each pairwise gcd by $(j - i) \log N$ . Since

\sum_{N - k + 1 \leq i < j \leq N} (j - i) = \sum_{h = 1}^{k - 1} h (k - h) = (\binom{k + 1}{3}),

taking logarithms gives (10). ◻

Theorem 3.4 (common-denominator growth).

\underset{N \to \infty}{lim inf} \frac{\log L_{N}}{N^{3 / 2} \log N} \geq \frac{2 \sqrt{2}}{3} .

Proof. The sum of the logarithms on a terminal block of length $k$ is of order $k N \log N$ , while the gcd estimate costs order $k^{3} \log N$ . We therefore fix $α > 0$ and take $k = ⌊ α \sqrt{N} ⌋$ . For large $N$ this is an admissible block length. If $u = N - k + 1$ , then $n! - 1 \geq n! / 2$ and $\log n! \geq n \log n - n$ give

\sum_{n = u}^{N} \log (n! - 1) \geq k (u \log u - u) - k \log 2.

Here $k = α \sqrt{N} + O (1)$ and $u = N + O (\sqrt{N})$ , so $k u \log u = α N^{3 / 2} \log N + O (N \log N)$ . The remaining two terms are $O (N^{3 / 2})$ . Also

(\binom{k + 1}{3}) \log N = \frac{α^{3}}{6} N^{3 / 2} \log N + o (N^{3 / 2} \log N) .

After division by $N^{3 / 2} \log N$ , Lemma 3.3 therefore yields a lower limit of at least $α - α^{3} / 6$ . This expression is maximised at $α = \sqrt{2}$ , giving $2 \sqrt{2} / 3$ . ◻

The displayed proof of the lcm bound uses only Lemmas 3.1 and 3.2. Theorem 12 of Garaev, Luca and Shparlinski [7] supplies a different input: a uniform $O (N^{2 / 3})$ bound for the number of solutions of $n! \equiv a (\mod p)$ , for fixed $a ≢ 0 (\mod p)$ , in an interval of length $N$ contained in $1 \leq n < p$ . The weaker lcm deduction from it is retained in §B.3, not used here. No exhaustive priority claim is made for the lcm estimate; its precise elementary derivation and the inherited subtraction are stated so that the claim can be assessed without relying on a search conclusion.

Remark (polynomial shifts). Fix $P \in Z [X] ∖ {0}$ . By Lemma 3 of Luca and Shparlinski [13], in the form given by Lai [14], there is $n_{0} \geq 2$ , depending only on $P$ , such that $n! + P (n) > 1$ for all $n \geq n_{0}$ and $P (n) (n + 1) \dots (n + h) \neq P (n + h)$ for all $n \geq n_{0}$ and $h \geq 1$ . The nonvanishing has a short direct proof. Beyond a cutoff, $P$ has constant nonzero sign and $P (n + 1) / P (n) \to 1$ , so $0 < P (n + 1) / P (n) < n + 1$ for every sufficiently large $n$ . Multiplying these inequalities gives

0 < \frac{P (n + h)}{P (n)} < \prod_{j = 1}^{h} (n + j) (h \geq 1) .

Thus the required subtraction is nonzero for every gap $h$ , not merely for each fixed $h$ after a separate cutoff. Increasing $n_{0}$ also makes $n! + P (n) > 1$ . The proof of Theorem 3.4 then gives

\underset{N \to \infty}{lim inf} \frac{\log lcm {n! + P (n) : n_{0} \leq n \leq N}}{N^{3 / 2} \log N} \geq \frac{2 \sqrt{2}}{3} .

For $n_{0} \leq i < j \leq N$ , multiplying $i! + P (i)$ by $j! / i!$ and subtracting $j! + P (j)$ gives

gcd (i! + P (i), j! + P (j)) | \frac{j!}{i!} P (i) - P (j),

which replaces Lemma 3.2. The right-hand side is nonzero by the choice of $n_{0}$ , and its absolute value is at most $C_{P} N^{j - i + \deg P}$ for a constant $C_{P}$ . Over a block of $k$ indices the pairwise loss therefore grows by $O_{P} (k^{2} \log N)$ , which is $O_{P} (N \log N)$ when $k = ⌊ α \sqrt{N} ⌋$ . Since $n! + P (n) \geq n! / 2$ for large $n$ , the lower estimate for the sum of logarithms is unchanged. The cutoff is needed, since for $P = - 2$ the factor at $n = 2$ vanishes. The hypothesis $P \neq 0$ is needed as well, since for $P = 0$ the least common multiple is $N!$ whatever the cutoff.

A non-polynomial comparison.

Consequences for the support parameter

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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