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

A complementary-denominator criterion

We seek a multiplier with two properties: it absorbs each fixed rational denominator eventually, and the scaled partial sum has a computable gap to the next integer. Include a factorial to ensure the first property. For the second, the multiplier also absorbs every prime-power level shared by two summand denominators. Any exponent left in the denominator then comes from a unique maximal occurrence, so it cannot cancel on reduction. This multiplier is sufficient, not necessarily minimal. For an integer $p \geq 3$ , not necessarily prime, put $I_{p} = {2, \dots, 2 p - 1}$ and $F_{p} = (p - 1)!$ , and define

\begin{aligned} D_{p} & = {lcm}_{\begin{matrix} i, j \in I_{p} \\ i < j \end{matrix}} gcd (d_{i}, d_{j}), & C_{p} & = lcm (F_{p}, D_{p}), \\ L_{p}^{blk} & = lcm (F_{p}, d_{2}, \dots, d_{2 p - 1}), & R_{p} & = L_{p}^{blk} / C_{p} . \end{aligned}

Prime by prime, $D_{p}$ records the second-largest exponent, $C_{p}$ raises it to at least the exponent in $F_{p}$ , and $R_{p}$ records what remains below the largest exponent. The factors $C_{p}$ and $R_{p}$ need not be coprime: a prime absent from $F_{p}$ with largest exponent $5$ and second-largest exponent $2$ occurs to powers $2$ and $3$ in the two factors. This is an illustration of the valuation calculation, not a claimed pattern in a particular factorial block. It will be useful to retain the ratio ${\tilde{C}}_{p} = C_{p} / F_{p}$ , so that

\begin{matrix} (23) & L_{p}^{blk} = C_{p} R_{p} = F_{p} {\tilde{C}}_{p} R_{p} . \end{matrix}

Including $F_{p}$ in both $C_{p}$ and $L_{p}^{blk}$ ensures that a fixed rational denominator divides $C_{p}$ for all large $p$ . The ratio ${\tilde{C}}_{p}$ removes precisely that factorial factor when we estimate the shared part. Put

T_{p} = \sum_{i \in I_{p}} \frac{L_{p}^{blk}}{d_{i}}, ρ_{p} = (- T_{p}) mod R_{p}, K_{p} = 2 p^{2} (2 p - 1)!,

with $ρ_{p}$ the least nonnegative representative. All of these depend only on a finite partial sum.

For each prime, $C_{p}$ contains the larger of its exponent in $F_{p}$ and its second-largest exponent among the denominators. A prime dividing $R_{p}$ therefore has a unique denominator whose exponent is maximal and exceeds the exponent in $F_{p}$ , so the argument of Theorem 2.1 applies to it and gives $gcd (T_{p}, R_{p}) = 1$ . Thus $R_{p} = den (C_{p} H_{2 p - 1})$ , the reduced denominator of the scaled partial sum, not generally of $H_{2 p - 1}$ . When $R_{p} > 1$ , this ratio is nonintegral and

\begin{matrix} (24) & \frac{ρ_{p}}{R_{p}} = ⌈ C_{p} H_{2 p - 1} ⌉ - C_{p} H_{2 p - 1} \in (0, 1), \end{matrix}

the distance to the next integer. When $R_{p} = 1$ , $ρ_{p} = 0$ but the scaled partial sum is integral, so the least strictly larger integer is at distance $1$ . The theorem below uses only $R_{p} > 1$ .

At $p = 3$ , the denominators $1, 5, 23, 119$ are pairwise coprime. Thus $D_{3} = 1$ , $C_{3} = 2$ , $R_{3} = 13685$ , and

C_{3} H_{5} = \frac{34264}{13685}, \frac{ρ_{3}}{R_{3}} = 3 - \frac{34264}{13685} = \frac{6791}{13685} .

The tail estimate in the proof below gives $C_{3} (S - H_{5}) < 7 / 1080 < 6791 / 13685$ . This verifies the comparison for one partial sum. The theorem needs this comparison at arbitrarily large $p$ , so that $F_{p}$ can absorb any fixed rational denominator.

Theorem 5.1 (a sufficient tail inequality). Suppose that for every $B$ there is a natural parameter $p \geq 3$ with $p > B$ , $R_{p} > 1$ , and

\begin{matrix} (25) & (2 p + 1) L_{p}^{blk} < K_{p} ρ_{p} . \end{matrix}

Then $S$ is irrational.

Proof. Suppose $S = a / q$ with $q \geq 1$ , and choose a parameter in the hypothesis with $p > q$ . Since $q ∣ F_{p} ∣ C_{p}$ , the number $C_{p} S$ is an integer strictly above $C_{p} H_{2 p - 1}$ . The gap identity (24) gives

C_{p} (S - H_{2 p - 1}) \geq \frac{ρ_{p}}{R_{p}} .

On the other hand, $1 / (n! - 1) < 2 / n!$ for $n \geq 2 p$ , and comparison with a geometric series of ratio $1 / (2 p + 1)$ gives

S - H_{2 p - 1} < \frac{2}{(2 p)!} \frac{2 p + 1}{2 p} = \frac{2 p + 1}{K_{p}} .

Multiplication by $C_{p}$ and (25), with $L_{p}^{blk} = C_{p} R_{p}$ , now give $C_{p} (S - H_{2 p - 1}) < ρ_{p} / R_{p}$ , a contradiction. ◻

Cancelling $R_{p}$ in (25) through (23) displays the quantitative issue directly:

\begin{matrix} (26) & (2 p + 1) C_{p} < K_{p} \frac{ρ_{p}}{R_{p}} . \end{matrix}

The factor $R_{p}$ has cancelled. Thus new prime divisors, even large ones, do not by themselves bound $C_{p}$ or keep $ρ_{p} / R_{p}$ away from zero. Proposition 2.2 supplies the primes but not these two estimates.

Under $R_{p} > 1$ , we have $ρ_{p} > 0$ , and taking logarithms gives

\log {\tilde{C}}_{p} - \log \frac{ρ_{p}}{R_{p}} < \log \frac{2 p^{2} (2 p - 1)!}{(2 p + 1) (p - 1)!} = p \log p + (2 \log 2 - 1) p + O (\log p) .

Both terms on the left must be controlled at the same $p$ . The elementary bound $ρ_{p} \geq 1$ gives only $ρ_{p} / R_{p} \geq 1 / R_{p}$ , which may be too small. For a concrete comparison, a gap of at least $1 / 2$ would reduce the sufficient test to $(2 p + 1) C_{p} < K_{p} / 2$ . At the opposite extreme, the gap $1 / R_{p}$ requires the stronger bound $(2 p + 1) C_{p} < K_{p} / R_{p}$ . Neither type of favourable behaviour is proved here for an unbounded family of these particular prefixes.

A more restrictive test also uses a prime $q$ that first divides $m! - 1$ at an index $m \geq 4$ . Choose $p = ⌊ m / 2 ⌋ + 1$ , which need not be prime. This choice puts $m$ in the block and gives $q ∣ R_{p}$ . Indeed, $q > m \geq p$ , so $q ∤ F_{p}$ , and no earlier denominator contains $q$ . The only possible later index in the block is $m + 1$ ; when it occurs, $(m + 1)! - 1 \equiv m ≢ 0 (\mod q)$ . Thus $q$ occurs in just one block denominator and is absent from $C_{p}$ .

The additional comparison uses the integer $R_{p} / q$ . Requiring the tail bound to be smaller than both $ρ_{p}$ and $R_{p} / q$ gives $min (ρ_{p}, R_{p} / q)$ and is equivalent to the following two inequalities:

\begin{matrix} (27) & (2 p + 1) L_{p}^{blk} < K_{p} min (ρ_{p}, R_{p} / q) ⟺ {\begin{cases} (2 p + 1) L_{p}^{blk} < K_{p} ρ_{p}, \\ (2 p + 1) C_{p} q < K_{p} . \end{cases} \end{matrix}

The equivalence follows by testing each entry of the minimum and using (23). Both inequalities must hold for the same arbitrarily large parameters. The first alone is already sufficient by Theorem 5.1; the second is an extra restriction, not an equivalent formulation of irrationality.

A fixed small index cannot supply new factors indefinitely. If $n$ is fixed and $p > n! - 1$ , then $n! - 1$ divides $(p - 1)!$ , so none of its prime powers remains outside the factorial base. For example, the factor $719$ of $6! - 1$ is useful only over a finite range of $p$ . Any argument using such factors at arbitrarily large $p$ must let their indices increase.

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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