Skip to content

Erdős problem 692

Start work

Let δ1(n,m)\delta_1(n,m) be the density of the set of integers with exactly one divisor in (n,m)(n,m). Is δ1(n,m)\delta_1(n,m) unimodular for m>n+1m>n+1 (i.e. increases until some mm then decreases thereafter)?

Sources

The source's own material, retained exactly.

5 retained statements36c6d0fef71a

Open selected source

FormalConjectures/ErdosProblems/

692.lean

Retained formal statement1 of 5

Let δ1(n,m)\delta_1(n,m) be the density of the set of integers with exactly one divisor in (n,m)(n,m). Is δ1(n,m)\delta_1(n,m) unimodular for m>n+1m>n+1 (i.e. increases until some mm then decreases thereafter)?

Cambie has calculated that unimodularity fails even for n=2n=2 and n=3n=3.

FormalConjectures/ErdosProblems/692.leanErdos692.erdos_692.parts.i1 lineExact file
False ↔ ∀ (δ : ℕ → ℕ → ℝ), (∀ (a b : ℕ), Erdos692.IsDelta₁ a b (δ a b)) → ∀ (n : ℕ), UnimodularOn (δ n) (n + 1)
SolvedStatement only, no proof