Is $n!$ a perturbation-robust irrationality sequence? ($2^n$ is now known to fail) (Erdős #264)
Statement
Call a sequence of positive integers $a_1,a_2,\ldots$ *perturbation-robust* if, for every bounded sequence of integers $b_n$ (with $b_n\neq 0$ and $a_n+b_n\neq 0$ for all $n$), the sum $$\sum_n \frac{1}{a_n+b_n}$$ is irrational. Are $a_n=2^n$ or $a_n=n!$ examples of such a sequence? (For $a_n=2^n$ the answer is now known to be no; the case $a_n=n!$ remains open — see background.)
Acceptance. FULLY RESOLVES: decide whether $a_n=n!$ is perturbation-robust — either a proof that $\sum_n 1/(n!+b_n)$ is irrational for every admissible bounded integer sequence $b_n$, or an explicit bounded $b_n$ (with $b_n\neq 0$, $n!+b_n\neq 0$) making the sum rational, with the rational value certified. A machine-checkable Lean 4 proof extending ErdosProblems/264.lean is the gold standard; otherwise a complete written proof with every step. ADVANCES: a partial or conditional result strictly beyond the background — e.g. prove irrationality of $\sum 1/(n!+b_n)$ for all $b_n$ in a restricted class (constant $b_n$, or all $|b_n|\le B$ for fixed $B$); or identify a new growth threshold guaranteeing or forbidding the property, improving on the '$\limsup a_{n+1}/a_n<\infty\Rightarrow$ fails' boundary, with proof. Deliver the Lean proof file or full manuscript, or the explicit rational-making perturbation.
Background
Posed by Erdős and Graham [ErGr80, p.63] and reiterated by Erdős in [Er88c, p.105]; listed as open on erdosproblems.com/264 (fetched 2026-07-13, status 'open', tagged 'irrationality'). This is one of several competing definitions of an 'irrationality sequence' (compare Erdős #262 and Erdős #263, erdosproblems.com/262 and /263); Erdős and Graham noted $a_n=2^{2^n}$ is an example. They asked whether such a sequence can have polynomial growth; Erdős retracted this in [Er88c], stating 'It is not hard to show that it cannot increase slower than exponentially'. Kovač and Tao [KoTa24] proved that $a_n=2^n$ is NOT perturbation-robust; more generally, any strictly increasing sequence with $\sum 1/a_n$ convergent and $\liminf(a_n^2\sum_{k>n}a_k^{-2})>0$ fails — in particular any strictly increasing sequence with $\limsup a_{n+1}/a_n<\infty$ fails. Conversely, they proved that for any function $F$ with $\lim F(n+1)/F(n)=\infty$ there exists a perturbation-robust sequence with $a_n\sim F(n)$. Since $n!$ has $a_{n+1}/a_n=n+1\to\infty$, the negative result does not apply to it, but no proof that $n!$ itself is perturbation-robust is known — that is the open core. The statement is formalised in Lean (DeepMind Formal Conjectures, ErdosProblems/264.lean). Attacker's tool: Diophantine irrationality/approximation methods for boundedly-perturbed reciprocal series with factorial denominators, ideally certified via the existing Lean 4 formalisation.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #264 (T. F. Bloom) | website |
| REF-02 | Lean formalisation of Erdős #264 (DeepMind Formal Conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.