[#P3008] Irrationality of the Factorial Reciprocal Minus One Series
Problem. Erdős Problem 68: Consider the infinite series \[ S = \sum_{n=2}^{\infty} \frac{1}{n! - 1}. \] Is \(S\) irrational?
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.
2Problem setup
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.
Remark 1. 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.
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
Current 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
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
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
2See also
How to cite
TheoremDB contributors, “Irrationality of the Factorial Reciprocal Minus One Series,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-68This page as plain text: erdos-problem-68.md
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.