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import FormalConjecturesUtilErdős Problem 315
References:
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Ka25] Y. Kamio, Asymptotic analysis of infinite decompositions of a unit fraction into unit fractions. arXiv:2503.02317 (2025).
[LiTa25] Z. Li and Q. Tang, On a conjecture of Erdős and Graham about the Sylvester's sequence. arXiv:2503.12277 (2025).
open Filternamespace Erdos315
The sequence $u_1=1$, $u_{n+1}=u_n(u_n+1)$, indexed here so that u i is $u_{i+1}$:
u 0 = 1, u 1 = 2, u 2 = 6, u 3 = 42, … (so u i + 1 is Sylvester's sequence
$2, 3, 7, 43, \ldots$).
def u : ℕ → ℕ
| 0 => 1
| n + 1 => u n * (u n + 1)
The Vardi constant $c_0=\lim u_n^{1/2^n}=1.264085\cdots$. With the 0-indexing of u, the
$n$-th term $u_n^{1/2^n}$ of the defining sequence is (u i : ℝ) ^ ((1 / 2 : ℝ) ^ (i + 1)) at
i = n - 1.
noncomputable def c₀ : ℝ := limUnder atTop fun i => (u i : ℝ) ^ ((1 / 2 : ℝ) ^ (i + 1))Let $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, so that $\sum_{k\geq 1}\frac{1}{u_k+1}$ and $u_k=\lfloor c_0^{2^k}+1\rfloor$ for $k\geq 1$, where $$c_0=\lim u_n^{1/2^n}=1.264085\cdots.$$ Let $a_1<a_2<\cdots $ be any other sequence with $\sum \frac{1}{a_k}=1$. Is it true that $$\liminf a_n^{1/2^n}<c_0=1.264085\cdots?$$
This is true, and was proved independently by Kamio [Ka25] and Li and Tang [LiTa25].
An earlier interpretation of this question on this site defined $u_1=2$ and $u_{n+1}=u_n^2-u_n+1$ (Sylvester's sequence), which is the same sequence shifted by $1$; we use the phrasing above as more faithful to [ErGr80]. The constant $c_0$ is called the Vardi constant.
@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos315.lean"]
theorem erdos_315 : answer(True) ↔
∀ a : ℕ → ℕ, (∀ i, 0 < a i) → StrictMono a → (∃ i, a i ≠ u i + 1) →
∑' i, (1 : ℝ) / a i = 1 →
atTop.liminf (fun i => (a i : ℝ) ^ ((1 / 2 : ℝ) ^ (i + 1))) < c₀ := ⊢ True ↔
∀ (a : ℕ → ℕ),
(∀ (i : ℕ), 0 < a i) →
StrictMono a →
(∃ i, a i ≠ u i + 1) → ∑' (i : ℕ), 1 / ↑(a i) = 1 → liminf (fun i ↦ ↑(a i) ^ (1 / 2) ^ (i + 1)) atTop < c₀
All goals completed! 🐙end Erdos315