1. FormalConjectures
  2. ErdosProblems
  3. «267»
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Erdős Problem 267

Reference: erdosproblems.com/267

namespace Erdos267

Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence. Let $n_1 < n_2 < \dots$ be an infinite sequence with $\frac{n_{k+1}}{n_k} \ge c > 1$. Must $\sum_k \frac 1 {F_{n_k}}$ be irrational?

@[category research open, AMS 11] theorem declaration uses 'sorry'erdos_267 : answer(sorry) ↔ ∀ᵉ (n : ℕ → ℕ) (c > (1 : ℚ)), StrictMono n → (∀ k, c ≤ n (k+1) / n k) → Irrational (∑' k, 1 / (Nat.fib <| n k)) := ⊢ True ↔ ∀ (n : ℕ → ℕ), ∀ c > 1, StrictMono n → (∀ (k : ℕ), c ≤ ↑(n (k + 1)) / ↑(n k)) → Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib (n k))) All goals completed! 🐙

Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence. Let $n_1 < n_2 < \dots$ be an infinite sequence with $\frac {n_k}{k} \to \infty$. Must $\sum_k \frac 1 {F_{n_k}}$ be irrational?

@[category research open, AMS 11] theorem declaration uses 'sorry'erdos_267.variants.generalisation_ratio_limit_to_infinity : answer(sorry) ↔ ∀ (n : ℕ → ℕ), StrictMono n → Filter.Tendsto (fun k => (n (k+1) / k.succ : ℝ)) Filter.atTop Filter.atTop → Irrational (∑' k, 1 / (Nat.fib <| n k)) := ⊢ True ↔ ∀ (n : ℕ → ℕ), StrictMono n → Filter.Tendsto (fun k => ↑(n (k + 1)) / ↑k.succ) Filter.atTop Filter.atTop → Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib (n k))) All goals completed! 🐙

Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\sum_n \frac 1 {F_{2^n}}$ is irrational.

Formal proof provided by AlphaProof Ref:

    [Go74] Good, I. J., A reciprocal series of Fibonacci numbers

    [BiHo76] Hoggatt, Jr., V. E. and Bicknell, Marjorie, A reciprocal series of Fibonacci numbers with subscripts $2\sp{n}k$

@[category research solved, AMS 11, formal_proof using formal_conjectures at "https://github.com/mo271/formal-conjectures/blob/2663234a28260853790aa5752d8d4550ff0ab1ca/FormalConjectures/ErdosProblems/267.lean#L56"] theorem declaration uses 'sorry'erdos_267.variants.specialization_pow_two : Irrational <| ∑' k, 1 / (Nat.fib <| 2^k) := ⊢ Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib (2 ^ k))) All goals completed! 🐙

The sum $\sum_n \frac 1 {F_{n}}$ itself was proved to be irrational by André-Jeannin.

Ref: André-Jeannin, Richard, Irrationalité de la somme des inverses de certaines suites récurrentes.

@[category research solved, AMS 11] theorem declaration uses 'sorry'erdos_267.variants.fibonacci_inverse_sum : Irrational <| ∑' k, 1 / (Nat.fib k) := ⊢ Irrational (∑' (k : ℕ), 1 / ↑(Nat.fib k)) All goals completed! 🐙 end Erdos267