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import FormalConjecturesUtilErdős Problem 269
namespace Erdos269
A positive integer $n$ has all its prime factors in the set $P$. By convention, $1$ satisfies this for any $P$ as it has no prime divisors.
def HasPrimeFactorsIn (P : Set ℕ) (n : ℕ) : Prop :=
n > 0 ∧ ∀ p, p.Prime → p ∣ n → p ∈ P
The infinite, strictly increasing sequence ${a_0, a_1, \dots}$ of integers whose prime factors all belong to $P$.
noncomputable def a (P : Set ℕ) : ℕ → ℕ := Nat.nth <| HasPrimeFactorsIn P
The $n$-th partial least common multiple, $[a_0, \dots, a_{n-1}]$, which is the LCM of the first $n$ integers in the sequence.
noncomputable def partialLcm (P : Set ℕ) (n : ℕ) : ℕ :=
-- We take the LCM of `{a P 0, ..., a P n}`.
(Finset.range n).lcm (a P)
The sum $\sum_{n=1}^\infty \frac{1}{[a_0,\ldots,a_{n - 1}]}$.
noncomputable def series (P : Set ℕ) : ℝ := ∑' n, (1 : ℝ) / (partialLcm P n)
Let $P$ be a finite set of primes with $|P| \ge 2$ and let ${a_1 < a_2 < \dots}$ be the set of positive integers whose prime factors are all in $P$. Is the sum $$ \sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]} $$ rational?
@[category research open, AMS 11]
theorem erdos_269.variants.rational : answer(sorry) ↔
∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),
∃ (q : ℚ), q = (series (P : Set ℕ)) := ⊢ True ↔ ∀ (P : Finset ℕ), (∀ p ∈ P, Nat.Prime p) → P.card ≥ 2 → ∃ q, ↑q = series ↑P
All goals completed! 🐙
Let $P$ be a finite set of primes with $|P| \ge 2$ and let ${a_1 < a_2 < \dots}$ be the set of positive integers whose prime factors are all in $P$. Is the sum $$ \sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]} $$ irrational?
@[category research open, AMS 11]
theorem erdos_269.variants.irrational : answer(sorry) ↔
∀ᵉ (P : Finset ℕ) (h : ∀ p ∈ P, p.Prime) (h_card : P.card ≥ 2),
Irrational (series (P : Set ℕ)) := ⊢ True ↔ ∀ (P : Finset ℕ), (∀ p ∈ P, Nat.Prime p) → P.card ≥ 2 → Irrational (series ↑P)
All goals completed! 🐙
This theorem addresses the case where the set of primes $P$ is infinite. In this case the sum is irrational.
@[category research solved, AMS 11]
theorem erdos_269.variants.infinite (P : Set ℕ) (h : ∀ p ∈ P, p.Prime) (h_inf : P.Infinite) :
Irrational (series P) := P:Set ℕh:∀ p ∈ P, Nat.Prime ph_inf:P.Infinite⊢ Irrational (series P)
All goals completed! 🐙
end Erdos269