[#P3008] Irrationality of the Factorial Reciprocal Minus One Series
Contents
Problem. Erdős Problem 68: Consider the infinite series \[ S = \sum_{n=2}^{\infty} \frac{1}{n! - 1}. \] Is \(S\) irrational?
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Definitions and notation
1Context
This problem asks whether a specific rapidly convergent series involving factorials yields an irrational number. The series can be rewritten using the geometric series formula as \[ \sum_{n=2}^{\infty} \frac{1}{n! - 1} = \sum_{n=2}^{\infty} \sum_{k=1}^{\infty} \frac{1}{(n!)^k}, \] since for each \(n \geq 2\), we have \(\frac{1}{n! - 1} = \sum_{k=1}^{\infty} (n!)^{-k}\) with \(n! > 1\) ensuring convergence.
2Definitions
Definition 1 (For a non-negative integer \(n\), the factorial \(n!\). For a non-negative integer \(n\), the factorial \(n!\) is the product of all positive integers up to \(n\), with \(0! = 1\).
Definition 2 (A real number). A real number is irrational if it cannot be expressed as a ratio of two integers.
Definition 3 (The notation \(\sum_{n=0}^{\infty} a_n\). The notation \(\sum_{n=0}^{\infty} a_n\) denotes the sum of an infinite series, defined as the limit of the partial sums when this limit exists.
3What counts as a solution
- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 68: Consider the infinite series \[ S = \sum_{n=2}^{\infty} \frac{1}{n! - 1}. \] Is \(S\) irrational?
1Status
Saved packet · July 31, 2026
Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 68 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 68: Consider the infinite series \[ S = \sum_{n=2}^{\infty} \frac{1}{n! - 1}. \] Is \(S\) irrational?[1]
1Packet records
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Notes and companion material
Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 68 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 68 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
- The pinned Formal Conjectures declaration was matched by problem number and source locator.
- The controlled TheoremDB source corpus was checked for an already published record with the same slug.
How the 2 records connect
ProblemIrrationality of the Factorial Reciprocal Minus One Series
All 1 recorded relations between these records and the problem
2See also
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Cite this problem statement
Cite the original sources separately.
“Irrationality of the Factorial Reciprocal Minus One Series.” TheoremDB. P3008. Problem statement; statement text SHA-256 69b110665ffe23d2af4cac010c8460f4552313ea9cfb75a2f219ae3cdda4e1b1. https://theoremdb.org/statement/?ref=P3008
@misc{theoremdb-problem-69b110665ffe23d2af4cac010c8460f4552313ea9cfb75a2f219ae3cdda4e1b1,
title = {{Irrationality of the Factorial Reciprocal Minus One Series}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 69b110665ffe23d2af4cac010c8460f4552313ea9cfb75a2f219ae3cdda4e1b1},
url = {https://theoremdb.org/statement/?ref=P3008}
}Plain text: Built Markdown snapshot
This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com[1], current as of July 31, 2026.
1References
- Packet source. Erdős Problems database, Problem 68, maintained status record. Erdős Problems record 68, checked 2026-08-01. Problem 68; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 68; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Irrationality of the Factorial Reciprocal Minus One Series: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
- Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 68 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Irrationality of the Factorial Reciprocal Minus One Series: Pins the maintained database snapshot used for this release's dated status decision.
- Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 68. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/68.lean:L33; theorem erdos_68; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Irrationality of the Factorial Reciprocal Minus One Series: Supplies the pinned formal declaration whose human-readable editorial statement is published here.
Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.
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