FormalConjectures.ErdosProblems.«252»

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import FormalConjecturesUtil
import FormalConjectures.Wikipedia.Schinzel

Erdős Problem 252

References:

erdosproblems.com/252

[ErSt71] Erdös, P., and E. G. Straus. "Some number theoretic results." Pacific J. Math 36 (1971): 635-646.

[ErSt74] Erdős, Paul, and Ernst Straus. "On the irrationality of certain series." Pacific journal of mathematics 55.1 (1974): 85-92.

[ErKa54] P. Erdős, M. Kac, Amer. Math. Monthly 61 (1954), Problem 4518.

[ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J. (2006), 455-460.

[FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).

[Pr22] Pratt, K., The irrationality of a divisor function series of Erdős and Kac. arXiv:2209.11124 (2022).


open scoped Nat ArithmeticFunction.sigma


namespace Erdos252

The series ∑ σ k n / n!.

noncomputable def erdos_252_sum (k : ℕ) : ℝ := ∑' n, σ k n / (n ! : ℝ)

Erdős Problem 252: irrationality of the sum for a given $k$.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_252 :
    answer(sorry) ↔ ∀ k ≥ 1, Irrational (erdos_252_sum k) := by⊢ True ↔ ∀ k ≥ 1, Irrational (erdos_252_sum k)
  sorryAll goals completed! 🐙

∑ σ 0 n / n! is irrational. This is proved in [ErSt71].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.k_eq_zero : Irrational (erdos_252_sum 0) := by⊢ Irrational (erdos_252_sum 0)
  sorryAll goals completed! 🐙

∑ σ 1 n / n! is irrational. This is proved in [ErSt74].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.k_eq_one : Irrational (erdos_252_sum 1) := by⊢ Irrational (erdos_252_sum 1)
  sorryAll goals completed! 🐙

∑ σ 2 n / n! is irrational. This is proved in [ErKa54].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.k_eq_two : Irrational (erdos_252_sum 2) := by⊢ Irrational (erdos_252_sum 2)
  sorryAll goals completed! 🐙

∑ σ 3 n / n! is irrational. This is proved in [ScPu06] and [FLC07].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.k_eq_three : Irrational (erdos_252_sum 3) := by⊢ Irrational (erdos_252_sum 3)
  sorryAll goals completed! 🐙

∑ σ 4 n / n! is irrational. This is proved in [Pr22].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.k_eq_four : Irrational (erdos_252_sum 4) := by⊢ Irrational (erdos_252_sum 4)
  sorryAll goals completed! 🐙

For a fixed k ≥ 5, is ∑ σ k n / n! irrational?.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.k_ge_five :
    answer(sorry) ↔ ∀ k ≥ 5, Irrational (erdos_252_sum k) := by⊢ True ↔ ∀ k ≥ 5, Irrational (erdos_252_sum k)
  sorryAll goals completed! 🐙

If Schinzel's conjecture is true, then ∑ σ k n / n! is irrational for all k. This is proved in [ScPu06].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.schinzel (hs : ∀ (fs : Finset (Polynomial ℤ)),
    (∀ f ∈ fs, BunyakovskyCondition f) → SchinzelCondition fs →
    Infinite ↑{n | ∀ f ∈ fs, Prime (Polynomial.eval (↑n) f).natAbs}) :
    ∀ k, Irrational (erdos_252_sum k) := byhs:∀ (fs : Finset (Polynomial ℤ)),
  (∀ f ∈ fs, BunyakovskyCondition f) →
    SchinzelCondition fs → Infinite ↑{n | ∀ f ∈ fs, Prime (Polynomial.eval n f).natAbs}⊢ ∀ (k : ℕ), Irrational (erdos_252_sum k)
  sorryAll goals completed! 🐙

If the prime k-tuples conjecture is true, then ∑ σ k n / n! is irrational. This is proved in [FLC07].

@[category research solved, AMS 11]
theorem declaration uses 'sorry'erdos_252.variants.prime_tuples {k : ℕ} (hk : 4 ≤ k) (hp : ∀ (a : Fin k → ℕ+)
    (b : Fin k → ℕ) (hab : ∀ p, p.Prime → ∃ n, ¬ p ∣ ∏ i, (a i * n + b i)),
    Set.Infinite {n | ∀ i : Fin k, (a i * n + b i).Prime} ) :
    Irrational (erdos_252_sum k) := byk:ℕhk:4 ≤ khp:∀ (a : Fin k → ℕ+) (b : Fin k → ℕ),
  (∀ (p : ℕ), Nat.Prime p → ∃ n, ¬p ∣ ∏ i, (↑(a i) * n + b i)) →
    {n | ∀ (i : Fin k), Nat.Prime (↑(a i) * n + b i)}.Infinite⊢ Irrational (erdos_252_sum k)
  sorryAll goals completed! 🐙


end Erdos252