# P3008: Irrationality of the Factorial Reciprocal Minus One Series - ID: `P3008` - Reference: `erdos-problem-68` - Page: https://theoremdb.org/statements/P3008 - Record maturity: Reviewed problem with recorded work ## Problem Erdős Problem 68: Consider the infinite series \[ S = \sum_{n=2}^{\infty} \frac{1}{n! - 1}. \] Is \(S\) irrational? ### Context 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. ### Problem setup - **Definition (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 (A real number).** A real number is irrational if it cannot be expressed as a ratio of two integers. - **Definition (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. - **Remark.** 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. ### What 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? ## 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. 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](#reference-1) ## Work ### Evidence for the current status **Claim 1 (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? 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. A complete resolution must satisfy this condition: 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? ### Background and intake notes - 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. ### Open directions - **Route 1** (reported): 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](#reference-1) ### Working on this Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-68`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome. ## References 1. Erdős Problems database, Problem 68, maintained status record. Erdős Problems record 68, checked 2026-08-01. Problem 68; status field and linked bibliography https://www.erdosproblems.com/68 - 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 - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source use: original_summary - Records the current open status and links the literature attached to this exact numbered problem. - 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. 2. Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 68 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31 - Source 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. 3. Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 68. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/68.lean:L33; theorem erdos_68; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/68.lean#L33 - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31 - Source 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.