Factorial Linear Forms and Denominators: Detailed Proofs and Rationality Criteria

Will Cook

Problem 0.1 (Erdős #68). Is $S = \sum_{n \geq 2} (n! - 1)^{- 1}$ irrational?

Erdős posed this question in 1988, together with the expectation that the corresponding series with denominators $n! + t$ should be transcendental for every integer $t$ [1]. For negative $t$ , the sum begins after all zero or negative denominators. Any further change of starting index adds a rational number. The expectation for all integer shifts remains unproved here. The problem number follows Bloom’s catalogue [2].

For finitely supported integer vectors on $n \geq 2$ , cancellation forms $M S + k$ give an explicitly computable ideal of multipliers. Correcting adjacent factorial differences at proper divisors gives an integral basis. Cancellation fixes its lower coordinates; excluding index one leaves a single Diophantine equation. Its gcd, computed below $2 D^{2}$ , gives the least positive multiplier and an attaining vector. At depth four, the support restriction strengthens $115 ∣ M$ to $1380 ∣ M$ . This classification is independent of nonintegrality.

The accompanying short paper proves this classification and bounds one resulting remainder between consecutive integers. Here we develop the support and coefficient-norm refinements and a primitive construction on arithmetic progressions. These solve different problems: at fixed $M$ , changing the vector changes $M S + k$ only by an integer, so it cannot alter nonintegrality. The progression construction makes $M$ divisible by every fixed denominator eventually, but still needs a gap estimate for those same forms.

The later sections study that arithmetic difficulty from two directions. Common-denominator growth measures the cost of clearing terms; prime-power calculations determine what survives reduction. Carries and factorial digits instead characterize rationality itself. The resulting equivalences identify what an irrationality proof must show, without proving that the required departures occur arbitrarily far out.

Throughout, write

d_{n} = n! - 1, H_{N} = \sum_{n = 2}^{N} \frac{1}{d_{n}}, L_{N} = lcm (d_{2}, \dots, d_{N}), A_{N} = \sum_{n = 2}^{N} \frac{L_{N}}{d_{n}} .

Thus $H_{N} = A_{N} / L_{N}$ . We write $den (x)$ for the positive reduced denominator of a rational number $x$ . Under $S = a / q$ , every form $M S + k$ with $M, k \in Z$ and $q ∣ M$ is integral. For each positive integer $q$ , an irrationality argument of this kind must produce a nonintegral form with $q ∣ M$ . Both properties must hold for the same vector; cancellation and growth of $M$ alone do not establish this.

The proof of the classification is in §1.1; the following table locates the additional arguments. In particular, the lower bound for $L_{N}$ concerns the unreduced common denominator, not automatically $den (H_{N})$ . The appendices retain the supplementary identities, limits of the attempted arguments, and formal-source concordance.

Question	Where it is treated
Which $M$ and coefficient vectors are possible?	§1.1
What do support and norm minimisation change?	§1.2
Which prime powers survive reduction?	§2
How fast does the common denominator grow?	§3
How does growth constrain the support parameter?	§3.1
Which eventual patterns characterize rationality?	§4
Which denominators are excluded by finite computations?	§6
Where are the formal declarations?	Appendix A.1