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import FormalConjecturesUtilErdős Problem 304
Reference: erdosproblems.com/304
open Asymptotics Filternamespace Erdos304
The set of k for which a / b can be expressed as a sum of k distinct unit fractions.
def unitFractionExpressible (a b : ℕ) : Set ℕ :=
{k | ∃ s : Finset ℕ, s.card = k ∧ (∀ n ∈ s, n > 1) ∧ (a / b : ℚ) = ∑ n ∈ s, (n : ℚ)⁻¹}@[category API, simp, AMS 11]
lemma zero_mem_unitFractionExpressible_iff {a b : ℕ} :
0 ∈ unitFractionExpressible a b ↔ a = 0 ∨ b = 0 := a:ℕb:ℕ⊢ 0 ∈ unitFractionExpressible a b ↔ a = 0 ∨ b = 0
All goals completed! 🐙a:ℕb:ℕh✝:a = 0 ∨ b = 0this:↑a / ↑b = 0s:Finset ℕhs:∀ n ∈ s, 1 < nh:∀ i ∈ s, i = 0⊢ ∀ (x : ℕ), x ∉ s
a:ℕb:ℕh✝:a = 0 ∨ b = 0this:↑a / ↑b = 0s:Finset ℕhs:∀ n ∈ s, 1 < nh:∀ i ∈ s, i = 0i:ℕhi:i ∈ s⊢ False
All goals completed! 🐙@[category API, AMS 11]
lemma unitFractionExpressible_zero_left {b : ℕ} :
unitFractionExpressible 0 b = {0} := unitFractionExpressible_of_zero (b:ℕ⊢ 0 = 0 ∨ b = 0 All goals completed! 🐙)@[category API, AMS 11]
lemma unitFractionExpressible_zero_right {a : ℕ} :
unitFractionExpressible a 0 = {0} := unitFractionExpressible_of_zero (a:ℕ⊢ a = 0 ∨ 0 = 0 All goals completed! 🐙)@[category API, AMS 11]
lemma zero_notMem_unitFractionExpressible {a b : ℕ} :
0 ∉ unitFractionExpressible a b ↔ a ≠ 0 ∧ b ≠ 0 := a:ℕb:ℕ⊢ 0 ∉ unitFractionExpressible a b ↔ a ≠ 0 ∧ b ≠ 0
All goals completed! 🐙@[category API, AMS 11]
lemma eq_inv_of_one_mem_unitFractionExpressible {a b : ℕ}
(h : 1 ∈ unitFractionExpressible a b) : ∃ m : ℕ, 1 < m ∧ (a / b : ℚ) = (m : ℚ)⁻¹ := a:ℕb:ℕh:1 ∈ unitFractionExpressible a b⊢ ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹
a:ℕb:ℕh:∃ s, (∃ a, s = {a}) ∧ (∀ n ∈ s, 1 < n) ∧ ↑a / ↑b = ∑ n ∈ s, (↑n)⁻¹⊢ ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹
a:ℕb:ℕm:ℕh₁:∀ n ∈ {m}, 1 < nh₂:↑a / ↑b = ∑ n ∈ {m}, (↑n)⁻¹⊢ ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹
a:ℕb:ℕm:ℕh₁:1 < mh₂:↑a / ↑b = (↑m)⁻¹⊢ ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹
All goals completed! 🐙a:ℕb:ℕh:1 ∈ unitFractionExpressible a bm:ℕhm₁:1 < mhm:↑a / ↑b = (↑m)⁻¹this:b ≠ 0⊢ a ∣ b
a:ℕb:ℕh:1 ∈ unitFractionExpressible a bm:ℕhm₁:1 < mhm:↑a / ↑b = (↑m)⁻¹this:b ≠ 0⊢ b = a * m
a:ℕb:ℕh:1 ∈ unitFractionExpressible a bm:ℕhm₁:1 < mthis:b ≠ 0hm:↑a * ↑m = ↑b⊢ b = a * m
All goals completed! 🐙
Let $$N(a, b)$$, denoted here by smallestCollection a b be the minimal k such that there
exist integers $1 < n_1 < n_2 < \dots < n_k$ with
$$\frac{a}{b} = \sum_{i=1}^k \frac{1}{n_i}$$
noncomputable def smallestCollection (a b : ℕ) : ℕ := sInf (unitFractionExpressible a b)a:ℕb:ℕha:a ≠ 0hb:b ≠ 0h:(unitFractionExpressible a b).Nonempty⊢ smallestCollection a b ≠ 0
a:ℕb:ℕha:a ≠ 0hb:b ≠ 0h:(unitFractionExpressible a b).Nonemptyh':smallestCollection a b = 0⊢ False
a:ℕb:ℕha:a ≠ 0hb:b ≠ 0h:(unitFractionExpressible a b).Nonemptyh':smallestCollection a b = 0this:0 ∈ unitFractionExpressible a b⊢ False
All goals completed! 🐙b:ℕhb:1 < bthis✝¹:1 ∈ unitFractionExpressible 1 bthis✝:smallestCollection 1 b ≤ 1this:0 ∉ unitFractionExpressible 1 b⊢ smallestCollection 1 b = 1
b:ℕhb:1 < bthis✝²:1 ∈ unitFractionExpressible 1 bthis✝¹:smallestCollection 1 b ≤ 1this✝:0 ∉ unitFractionExpressible 1 bthis:smallestCollection 1 b ≠ 0⊢ smallestCollection 1 b = 1
All goals completed! 🐙@[category API, AMS 11]
lemma eq_one_of_smallestCollection_eq_one {a b : ℕ}
(h : smallestCollection a b = 1) : ∃ m : ℕ, 1 < m ∧ (a / b : ℚ) = (m : ℚ)⁻¹ := a:ℕb:ℕh:smallestCollection a b = 1⊢ ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹
a:ℕb:ℕh:smallestCollection a b = 1this:1 ∈ unitFractionExpressible a b⊢ ∃ m, 1 < m ∧ ↑a / ↑b = (↑m)⁻¹
All goals completed! 🐙@[category API, AMS 11]
lemma dvd_of_smallestCollection_eq_one {a b : ℕ}
(h : smallestCollection a b = 1) : a ∣ b := a:ℕb:ℕh:smallestCollection a b = 1⊢ a ∣ b
a:ℕb:ℕh:smallestCollection a b = 1this:1 ∈ unitFractionExpressible a b⊢ a ∣ b
All goals completed! 🐙h:2 ∈ unitFractionExpressible 2 15this✝¹:smallestCollection 2 15 ≤ 2this✝:0 < smallestCollection 2 15this:smallestCollection 2 15 ≠ 1⊢ smallestCollection 2 15 = 2
All goals completed! 🐙Write $$N(b) = max_{1 \leq a < b} N(a, b)$$.
noncomputable def smallestCollectionTo (b : ℕ) : ℕ :=
sSup {smallestCollection a b | a ∈ Finset.Ico 1 b}In 1950, Erdős [Er50c] proved the upper bound $$N(b) \ll \log b / \log \log b$$. [Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg'{E}sz sz'{A}m'{u} megold'{A}sairól. Mat. Lapok (1950), 192-210.
@[category research solved, AMS 11]
theorem erdos_304.variants.upper_1950 :
(fun b => (smallestCollectionTo b : ℝ)) =O[atTop]
(fun b => Real.log b / Real.log (Real.log b)) := ⊢ (fun b ↦ ↑(smallestCollectionTo b)) =O[atTop] fun b ↦ Real.log ↑b / Real.log (Real.log ↑b)
All goals completed! 🐙In 1950, Erdős [Er50c] proved the lower bound $$\log \log b \ll N(b)$$. [Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg'{E}sz sz'{A}m'{u} megold'{A}sairól. Mat. Lapok (1950), 192-210.
@[category research solved, AMS 11]
theorem erdos_304.variants.lower_1950 :
(fun b : ℕ => Real.log (Real.log b)) =O[atTop]
(fun b => (smallestCollectionTo b : ℝ)) := ⊢ (fun b ↦ Real.log (Real.log ↑b)) =O[atTop] fun b ↦ ↑(smallestCollectionTo b)
All goals completed! 🐙In 1985 Vose [Vo85] proved the upper bound $$N(b) \ll \sqrt{\log b}$$. [Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.
@[category research solved, AMS 11]
theorem erdos_304.variants.upper_1985 :
(fun b => (smallestCollectionTo b : ℝ)) =O[atTop]
(fun b => Real.sqrt (Real.log b)) := ⊢ (fun b ↦ ↑(smallestCollectionTo b)) =O[atTop] fun b ↦ √(Real.log ↑b)
All goals completed! 🐙Is it true that $$N(b) \ll \log \log b$$?
@[category research open, AMS 11]
theorem upper_bound : answer(sorry) ↔
(fun b : ℕ => (smallestCollectionTo b : ℝ)) =O[atTop] (fun b : ℕ => Real.log (Real.log b)) := ⊢ True ↔ (fun b ↦ ↑(smallestCollectionTo b)) =O[atTop] fun b ↦ Real.log (Real.log ↑b)
All goals completed! 🐙end Erdos304