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

PDFOpen the PDF, 51 pages TeXLaTeX source Source on GitHub Problem page

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

Prime powers in reduced denominators

Let $p$ be a prime, and let $v_{p} (a)$ denote the exponent of $p$ in a positive integer $a$ . We determine which prime powers of $L_{M}$ survive in $den (H_{M})$ . At the largest exponent, this follows by specialising Louwsma and Martino’s valuation formula for an elementary symmetric sum [4]. Indeed, $\sum_{i} 1 / x_{i} = (\sum_{i} \prod_{j \neq i} x_{j}) / \prod_{i} x_{i}$ for positive integers $x_{i}$ . Their lemma evaluates the numerator’s valuation; subtracting the product’s valuation gives the reciprocal-sum form. The proof below uses the least common multiple instead of the product.

The distinction is already visible in $1 / 3 + 1 / 15 = 2 / 5$ : the common denominator contains $3$ , but the reduced denominator does not. By contrast, $1 / 9 + 1 / 45 = 2 / 15$ only lowers the exponent of $3$ from $2$ to $1$ . The theorem below tests whether the largest exponent is preserved; a failure of that test need not remove the prime completely. The formula is standard. Its mechanism is simple: after clearing by the lcm and reducing modulo $p$ , only denominators carrying the largest power of $p$ contribute. The factorial examples and later tail comparison concern this particular series.

Theorem 2.1 (maximal prime-power survival). Let $M \geq 2$ , let $p$ be a prime dividing $L_{M}$ , and put $e = v_{p} (L_{M})$ . Let $J = {n : 2 \leq n \leq M, v_{p} (d_{n}) = e}$ and write $d_{n} = p^{e} u_{n}$ for $n \in J$ . Then, with inverses in $F_{p}$ ,

\begin{matrix} (8) & v_{p} (den (H_{M})) = e ⟺ \sum_{n \in J} u_{n}^{- 1} \neq 0 in F_{p} . \end{matrix}

Proof. Write $L_{M} = p^{e} W$ with $p ∤ W$ . If $v_{p} (d_{n}) < e$ , then $p ∣ L_{M} / d_{n}$ . If $n \in J$ , then $u_{n} (L_{M} / d_{n}) = W$ , so $L_{M} / d_{n} \equiv W u_{n}^{- 1} (\mod p)$ . Summing over $2 \leq n \leq M$ gives

\begin{matrix} (9) & A_{M} \equiv \frac{L_{M}}{p^{e}} \sum_{n \in J} u_{n}^{- 1} (\mod p) . \end{matrix}

Finally $den (H_{M}) = L_{M} / gcd (A_{M}, L_{M})$ , whose $p$ -valuation is $e$ exactly when $p ∤ A_{M}$ . Since $W$ is invertible modulo $p$ , that is exactly the stated condition. ◻

The complete valuation identity is

v_{p} (den (H_{M})) = max {0, e - v_{p} (A_{M})} .

A zero residue in (8) therefore lowers the exponent, whereas complete cancellation requires $v_{p} (A_{M}) \geq e$ . For $e > 1$ , reduction modulo $p$ alone cannot distinguish partial from complete cancellation. The proof uses no property of factorials beyond the displayed denominators, so it applies to any finite sum of reciprocals of positive integers. For $d_{n} = n! - 1$ , the examples below show that complete cancellation can actually occur; the general valuation formula alone does not predict it.

Two complete cancellations.

A unique maximal exponent makes the residue sum a single nonzero term, so the condition holds automatically. With several maximal terms it can fail. The following examples show cancellation in the factorial sequence itself, not just in arbitrary rational sums. Take $M = p - 1$ . The table lists every index with $p ∣ d_{n}$ , and every one has valuation exactly one.

$p$	indices $n$	$d_{n} / p (\mod p)$	inverses modulo $p$
$139$	$69, 122, 137$	$6, 49, 73$	$116, 122, 40$
$2593$	$349, 2243, 2591$	$1508, 1566, 1678$	$1367, 356, 870$

The inverse sums are $278 = 2 \cdot 139$ and $2593$ . By Theorem 2.1, $139 ∤ den (H_{138})$ and $2593 ∤ den (H_{2592})$ : each prime divides the common denominator of its partial sum and vanishes from its reduced denominator. The listed indices and their valuations are checked by the recurrence

r_{1} = 1, r_{n} \equiv n r_{n - 1} (\mod p^{2}), 0 \leq r_{n} < p^{2},

run through $n = p - 1$ . At an index with $r_{n} \equiv 1 (\mod p)$ , the cofactor is $(r_{n} - 1) / p$ modulo $p$ . Valuation at least two would give $r_{n} = 1$ , which occurs for neither prime among the tested indices $2 \leq n \leq p - 1$ . That enumeration is a finite calculation separate from the theorem.

The recurrence is run modulo $p^{2}$ , not merely modulo $p$ : reduction modulo $p$ would find the indices with $p ∣ n! - 1$ , but would not determine whether $p^{2} ∣ n! - 1$ . This is why the zero inverse sums prove complete cancellation in these examples.

For each prime $p \geq 3$ , the exponent $v_{p} (den (H_{M}))$ is constant for $M \geq max (2, p - 2)$ . Wilson’s theorem gives $(p - 1)! - 1 \equiv - 2 (\mod p)$ , and $n! - 1 \equiv - 1 (\mod p)$ for $n \geq p$ . Hence every summand after index $p - 2$ has denominator coprime to $p$ . If a reduced fraction $a / b$ is followed by a summand $1 / d$ with $p ∤ d$ , their sum has numerator $a d + b$ over $b d$ . When $p ∣ b$ , this numerator is nonzero modulo $p$ , so the exponent of $p$ in the reduced denominator stays the same. When $p ∤ b$ , the new denominator remains coprime to $p$ . Consequently, the two cancellations persist in every later partial sum:

139 ∤ den (H_{M}) (M \geq 137), 2593 ∤ den (H_{M}) (M \geq 2591) .

The primes still divide the corresponding common denominators $L_{M}$ . These are assertions about the partial sums; they do not constrain a hypothetical denominator of their real limit $S$ without a further approximation estimate.

We next ask where a prime first divides a denominator $d_{m}$ . This means it divides none of $d_{2}, \dots, d_{m - 1}$ ; it may still divide a later denominator, so uniqueness at its first occurrence gives no uniqueness in every larger block.

Proposition 2.2 (cofinal first prime occurrences). For every integer $B \geq 0$ there are a prime $q$ and an integer $m > B$ with $m < q$ , $q ∣ m! - 1$ and $gcd (q, k! - 1) = 1$ for every $k$ with $2 \leq k < m$ .

Proof. Choose a prime $q \geq B! + 5$ . Wilson’s theorem gives $(q - 2)! \equiv 1 (\mod q)$ , so there is a least index $m \geq 2$ with $q ∣ m! - 1$ , and $m \leq q - 2$ . If $m \leq B$ , then $q \leq m! - 1 \leq B! - 1$ , contradicting the choice of $q$ . Hence $m > B$ , and minimality gives the coprimality assertions. ◻

The construction proves cofinality of first occurrences, but not the inequality involving $q$ needed for the additional two-factor test in §5. The sufficient tail inequality there does not require choosing a prime $q$ . Wilson reflection limits what can be inferred from a large prime factor alone: for odd $2 \leq n < q$ with $q ∣ n! - 1$ , the identity $(q - 1 - n)! n! \equiv (- 1)^{n + 1} (\mod q)$ gives the reflected congruence at $q - n - 1$ . It gives a second summand denominator only when $q \neq 2 n + 1$ and $n \leq q - 3$ . At the Wilson endpoint $n = q - 2$ , it instead returns index $1$ , where $1! - 1 = 0$ is not a denominator of this series. The reflection identity is classical; see Stewart [8].

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

Prefer the manuscript?

PDFOpen the PDF, 51 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:0af0d85ec1beab46. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.