893.lean`

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3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 893

References:

erdosproblems.com/893
[KoLu25] V. Kovač and F. Luca, On the number of divisors of Mersenne numbers. arXiv:2506.04883 (2025).

Imports (1)

FormalConjecturesUtil

def Erdos893.f Definition of function $f(n) := \sum_{1\leq k\leq n}\tau(2^k-1)$ line 39 source
Source documentation
Definition of function $f(n) := \sum_{1\leq k\leq n}\tau(2^k-1)$. Here $\tau$ is the divisor counting function, which is σ 0 in mathlib.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.Icc, OfNat.ofNat, ArithmeticFunction, ArithmeticFunction.sigma, HSub.hSub, HPow.hPow
14 instances, coercions and auxiliary constants left out.
```
def f (n : ℕ) : ℕ :=  ∑ k ∈ Finset.Icc 1 n, σ 0 (2^k - 1)
…
```
theorem Erdos893.erdos_893 Does the limit $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ tend to infinity? sorry line 47 source
Source documentation
Does the limit $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ tend to infinity?
(Other finite limits have been ruled out by [KoLu25], see below)

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos893.f, HMul.hMul, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_893 :
    answer(sorry) ↔ Tendsto (fun n : ℕ => (f (2 * n) : ℝ) / (f n : ℝ)) atTop atTop := by
…
```
Mentions: Erdos893.f
theorem Erdos893.erdos_893.variants.unbounded Documented declaration: unbounded sorry line 58 source
Source documentation
Kovač and Luca [KoLu25] (building on a heuristic independently found by Cambie (personal communication)) have shown that there is no finite limit, in that $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ is unbounded.

depends on sorry what this means
Statement uses: Not, BddAbove, Real, Set.range, Nat, HDiv.hDiv, Erdos893.f, HMul.hMul, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_893.variants.unbounded :
   ¬ BddAbove (Set.range fun n : ℕ => (f (2 * n) : ℝ) / f n) := by
…
```
Mentions: Erdos893.f