fact stringlengths 28 3.53k | type stringclasses 8
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values | imports listlengths 1 7 | filename stringlengths 33 87 | symbolic_name stringlengths 1 87 | docstring stringlengths 17 494 ⌀ |
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IsBehrend {ι : Type*} (A : ι → ℕ) : Prop := (MultiplesOf A).HasDensity 1 /-- A sequence of naturals $(a_i)$ is _weakly Behrend_ with respect to $\varepsilon \in \mathbb{R}$ if at least $1 - \varepsilon$ density of all numbers are a multiple of $A$. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | IsBehrend | /-- A sequence of naturals $(a_i)$ is _Behrend_ if almost all integers are a multiple of
some $a_i$. In other words, if the set of multiples has natural density $1$. -/ |
IsWeaklyBehrend {ι : Type*} (A : ι → ℕ) (ε : ℝ) : Prop := 1 - ε ≤ (MultiplesOf A).lowerDensity @[category test, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | IsWeaklyBehrend | /-- A sequence of naturals $(a_i)$ is _weakly Behrend_ with respect to $\varepsilon \in \mathbb{R}$
if at least $1 - \varepsilon$ density of all numbers are a multiple of $A$. -/ |
isBehrend_of_contains_one {ι : Type*} (A : ι → ℕ) (h : 1 ∈ Set.range A) : IsBehrend A := by rw [IsBehrend, Set.HasDensity] exact tendsto_atTop_of_eventually_const (i₀ := 1) fun n hn ↦ by field_simp [multiplesOf_eq_univ A h, Set.partialDensity] @[category test, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | isBehrend_of_contains_one | /-- A sequence of naturals $(a_i)$ is _weakly Behrend_ with respect to $\varepsilon \in \mathbb{R}$
if at least $1 - \varepsilon$ density of all numbers are a multiple of $A$. -/ |
isWeaklyBehrend_of_ge_one {ι : Type*} (A : ι → ℕ) {ε : ℝ} (hε : 1 ≤ ε) : IsWeaklyBehrend A ε := by exact (sub_nonpos.2 hε).trans (Set.lowerDensity_nonneg _) @[category test, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | isWeaklyBehrend_of_ge_one | null |
not_isWeaklyBehrend_of_neg {ι : Type*} (A : ι → ℕ) {ε : ℝ} (hε : ε < 0) : ¬IsWeaklyBehrend A ε := by norm_num [IsWeaklyBehrend] exact (add_lt_of_neg_right _ hε).trans_le (Set.lowerDensity_le_one _) /-- Let $A\subset\mathbb{N}$ be infinite such that $\sum_{a \in A} \frac{1}{a} = \infty$. Must there exist some $k\geq 1$ ... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | not_isWeaklyBehrend_of_neg | null |
erdos_26 : answer(sorry) ↔ ∀ A : ℕ → ℕ, StrictMono A → IsThick A → ∃ k, IsBehrend (A · + k) := by sorry /-- If we allow for $\sum_{a\in A} \frac{1}{a} < \infty$ then Rusza has found a counter-example. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | erdos_26 | /--
Let $A\subset\mathbb{N}$ be infinite such that $\sum_{a \in A} \frac{1}{a} = \infty$. Must
there exist some $k\geq 1$ such that almost all integers have a divisor of the form $a+k$
for some $a\in A$?
-/ |
erdos_26.variants.rusza : ∃ A : ℕ → ℕ, StrictMono A ∧ ¬IsThick A ∧ ∀ k, ¬IsBehrend (A · + k) := by sorry /-- Tenenbaum asked the weaker variant where for every $\epsilon>0$ there is some $k=k(\epsilon)$ such that at least $1-\epsilon$ density of all integers have a divisor of the form $a+k$ for some $a\in A$. -/ @[cate... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | erdos_26.variants.rusza | /--
If we allow for $\sum_{a\in A} \frac{1}{a} < \infty$ then Rusza has found a counter-example.
-/ |
erdos_26.variants.tenenbaum : answer(sorry) ↔ ∀ᵉ (A : ℕ → ℕ), StrictMono A → IsThick A → (∀ ε > (0 : ℝ), ∃ k, IsWeaklyBehrend (A · + k) ε) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/26.lean | erdos_26.variants.tenenbaum | /--
Tenenbaum asked the weaker variant where for every $\epsilon>0$ there is
some $k=k(\epsilon)$ such that at least $1-\epsilon$ density of all integers have a
divisor of the form $a+k$ for some $a\in A$.
-/ |
IsIrrationalitySequence (a : ℕ → ℕ) : Prop := ∀ b : ℕ → ℕ, BddAbove (Set.range b) → 0 ∉ Set.range (a + b) → 0 ∉ Set.range b → Irrational (∑' n, (1 : ℝ) / (a n + b n)) /-- Is $2^n$ an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24] [KoTa24] Kovač, V. and Tao T., On several irrationalit... | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/264.lean | IsIrrationalitySequence | /--
A sequence $a_n$ of integers is called an irrationality sequence if for every bounded sequence of integers $b_n$ with $a_n + b_n \neq 0$ and
$b_n \neq 0$ for all $n$, the sum
$$
\sum \frac{1}{a_n + b_n}
$$
is irrational.
Note: there are other possible definitions of this concept.
-/ |
erdos_264.parts.i : ¬IsIrrationalitySequence (2 ^ ·) := by sorry /-- Is $n!$ an example of an irrationality sequence? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/264.lean | erdos_264.parts.i | /--
Is $2^n$ an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24]
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
-/ |
erdos_264.parts.ii : answer(sorry) ↔ IsIrrationalitySequence Nat.factorial := by sorry /-- One example is $2^{2^n}$. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/264.lean | erdos_264.parts.ii | /--
Is $n!$ an example of an irrationality sequence?
-/ |
erdos_264.variants.example : IsIrrationalitySequence (fun n ↦ 2 ^ (2 ^ n)) := by sorry /-- Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers $a_n$ such that $\sum \frac{1}{a_n}$ converges and $$ \liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0 $$ is not an... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/264.lean | erdos_264.variants.example | /--
One example is $2^{2^n}$.
-/ |
erdos_264.variants.ko_tao_neg {a : ℕ → ℕ} (h₁ : StrictMono a) (h₂ : 0 ∉ Set.range a) (h₃ : Summable ((1 : ℝ) / a ·)) (h₄ : 0 < atTop.liminf fun n ↦ a n ^ 2 * ∑' k : Set.Ioi n, (1 : ℝ) / a k ^ 2) : ¬IsIrrationalitySequence a := by sorry /-- On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ wit... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/264.lean | erdos_264.variants.ko_tao_neg | /--
Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers
$a_n$ such that $\sum \frac{1}{a_n}$ converges and
$$
\liminf_{n \to \infty} (a_n^2 \sum_{k > n} \frac{1}{a_k^2}) > 0
$$
is not an irrationality sequence.
[KoTa24] Kovač, V. and Tao T., On several irrationality pro... |
erdos_264.variants.ko_tao_pos {F : ℕ → ℕ} (hF : atTop.Tendsto (fun n ↦ (F (n + 1) : ℝ) / F n) atTop) : ∃ a : ℕ → ℕ, IsIrrationalitySequence a ∧ (fun n ↦ (a n : ℝ)) ~[atTop] fun n ↦ (F n : ℝ) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/264.lean | erdos_264.variants.ko_tao_pos | /--
On the other hand, Kovač and Tao [KoTa24] do prove that for any function $F$ with
$\lim_{n \to \infty} \frac{F(n + 1)}{F(n)} = \infty$ there exists such an irrationality sequence with $a_n \sim F(n)$.
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series. arXiv:2406.17593 (2024).
-/ |
erdos_266 : ¬ ∀ (a : ℕ → ℕ), ((∀ n : ℕ, a n ≥ 1) ∧ Summable ((1 : ℝ) / a ·) → ∃ t ≥ (1 : ℕ), Irrational <| ∑' n, (1 : ℝ) / ((a n) + t)) := by sorry /-- In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing sequence $a_n$ of positive integers such that $\sum \frac{1}{a_n + t}$ converges to a ... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/266.lean | erdos_266 | /--
Let $a_n$ be an infinite sequence of positive integers such that $\sum \frac{1}{a_n}$ converges.
There exists some integer $t \ge 1$ such that $\sum \frac{1}{a_n + t}$ is irrational.
This was disproven by Kovač and Tao in [KoTa24].
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for Ahmes series.... |
erdos_266.variants.all_rationals : ∃ a : ℕ → ℕ, StrictMono a ∧ a 0 ≥ 1 ∧ (∀ t : ℚ, (¬ ∃ n : ℕ, t = -(a n)) → (∃ q : ℚ, HasSum (fun n : ℕ => ((1 : ℝ) / ((a n) + t))) q)) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/266.lean | erdos_266.variants.all_rationals | /--
In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing
sequence $a_n$ of positive integers such that $\sum \frac{1}{a_n + t}$ converges to a rational
number for all $t \in \mathbb{Q}$ such that $t \ne -a_n$ for any $n$.
[KoTa24] Kovač, V. and Tao T., On several irrationality problems for... |
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)) := by sorry /-- 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 $\su... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/267.lean | erdos_267 | /--
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?
-/ |
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)) := by sorry /-- Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\sum_n \frac 1 {F_... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/267.lean | erdos_267.variants.generalisation_ratio_limit_to_infinity | /--
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?
-/ |
erdos_267.variants.specialization_pow_two : Irrational <| ∑' k, 1 / (Nat.fib <| 2^k) := by sorry /-- 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,... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/267.lean | erdos_267.variants.specialization_pow_two | /--
Good [Go74] and Bicknell and Hoggatt [BiHo76] have shown that $\sum_n \frac 1 {F_{2^n}}$ is irrational.
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 $2\sp{n}k$_
-/ |
erdos_267.variants.fibonacci_inverse_sum : Irrational <| ∑' k, 1 / (Nat.fib k) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/267.lean | erdos_267.variants.fibonacci_inverse_sum | /--
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_.
-/ |
erdos_268 (d : ℕ) : (interior {x : Fin d → ℝ | ∃ A : Set ℕ, A.Infinite ∧ Summable (fun n : A => (1 : ℝ) / n ) ∧ x = fun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i)}).Nonempty := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/268.lean | erdos_268 | /-- Let `X` be the set of points in `Fin d → ℝ` of the shape
`fun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i)` for some infinite subset `A ⊆ ℕ` such that
`1 / n` is summable over `A`. `X` has nonempty interior. This is proved in [KoTa24].
-/ |
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 $P$. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/269.lean | HasPrimeFactorsIn | /--
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.
-/ |
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. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/269.lean | a | /--
The infinite, strictly increasing sequence $\{a_0, a_1, \dots\}$ of integers
whose prime factors all belong to $P$.
-/ |
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 $\sum_{n=1}^\infty \frac{1}{[a_0,\ldots,a_{n - 1}]}$. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/269.lean | partialLcm | /--
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.
-/ |
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 1... | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/269.lean | series | /--
The sum $\sum_{n=1}^\infty \frac{1}{[a_0,\ldots,a_{n - 1}]}$.
-/ |
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)) := by sorry /-- 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 ... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/269.lean | erdos_269.variants.rational | /--
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?
-/ |
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)) := by sorry /-- 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 | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/269.lean | erdos_269.variants.irrational | /--
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?
-/ |
erdos_269.variant.infinite (P : Set ℕ) (h : ∀ p ∈ P, p.Prime) (h_inf : P.Infinite) : Irrational (series P) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/269.lean | erdos_269.variant.infinite | /--
This theorem addresses the case where the set of primes $P$ is infinite. In this case the sum is
irrational.
-/ |
erdos_273 : answer(sorry) ↔ ∃ c : StrictCoveringSystem ℤ, ∀ i, ∃ (p : ℕ), p.Prime ∧ 5 ≤ p ∧ c.moduli i = Ideal.span {↑(p - 1)} := by sorry /-- Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 3$? -/ @[category research solved, AMS 5 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/273.lean | erdos_273 | /--
Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 5$?
-/ |
erdos_273.variants.three : answer(True) ↔ ∃ c : StrictCoveringSystem ℕ, ∀ i, ∃ p, p.Prime ∧ 3 ≤ p ∧ c.moduli i = Ideal.span {↑(p - 1)} := by -- TODO(Paul-Lez): find reference for this and perhaps formalize the proof? sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/273.lean | erdos_273.variants.three | /--
Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 3$?
-/ |
Group.ExactCovering (G : Type*) [Group G] (ι : Type*) [Fintype ι] where parts : ι → Subgroup G reps : ι → G nonempty (i : ι) : (parts i : Set G).Nonempty disjoint : (Set.univ (α := ι)).PairwiseDisjoint fun (i : ι) ↦ reps i • (parts i : Set G) covers : ⋃ i, reps i • (parts i : Set G) = Set.univ /-- If `G` is a group the... | structure | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/274.lean | Group.ExactCovering | null |
erdos_274 : answer(sorry) ↔ ∀ᵉ (G : Type*) (h : Group G) (hG : 1 < ENat.card G), (∃ (ι : Type*) (_ : Fintype ι) (P : Group.ExactCovering G ι), 1 < Fintype.card ι ∧ (Set.range P.parts).Pairwise fun A B ↦ #A ≠ #B) := by sorry /-- If `G` is a finite abelian group then there cannot exist an exact covering of `G` by more th... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/274.lean | erdos_274 | /--
If `G` is a group then can there exist an exact covering of `G` by more than one cosets of
different sizes? (i.e. each element is contained in exactly one of the cosets.)
-/ |
erdos_274.variants.abelian {G : Type*} [Fintype G] [CommGroup G] (hG : 1 < Fintype.card G) {ι : Type*} [Fintype ι] (P : Group.ExactCovering G ι) (hι : 1 < Fintype.card ι) : ∃ i j, i ≠ j ∧ #(P.parts i) = #(P.parts j) := by sorry /-- Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ be a finite system of left c... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/274.lean | erdos_274.variants.abelian | /--
If `G` is a finite abelian group then there cannot exist an exact covering of `G` by more
than one cosets of different sizes? (i.e. each element is contained in exactly one
of the cosets.)
-/ |
herzog_schonheim {G : Type*} [Group G] (hG : 1 < ENat.card G) {ι : Type*} [Fintype ι] (hι : 1 < Fintype.card ι) (P : Group.ExactCovering G ι) : ∃ i j, i ≠ j ∧ (P.parts i).index = (P.parts j).index := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/274.lean | herzog_schonheim | /--
Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ be a finite system of left cosets of
subgroups $G_1, \dots, G_k$ of $G$.
Herzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$, then the
indices $[G:G_1], \dots, [G:G_k]$ cannot be distinct.
-/ |
IsLucasSequence (L : ℕ → ℕ) : Prop := ∀ n, L (n + 2) = L (n + 1) + L n namespace Erdos276 /-- Is there an infinite Lucas sequence $a_0, a_1, \ldots$ where $a_{n+2} = a_{n+1} + a_n$ for $n \ge 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every term of the sequence? -/ @[category rese... | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/276.lean | IsLucasSequence | /--
We define a Lucas sequence to be a Fibonacci sequence with arbitrary starting points
`L 0` and `L 1`.
TODO: There seems to be multiple definitions in the literature, some of which also
allow coefficients in the reccurence relation. For now this simple definition has been
chosen as it agrees best with the Erdős pro... |
erdos_276 : answer(sorry) ↔ ∃ (a : ℕ → ℕ), IsLucasSequence a ∧ (∀ k, (a k).Composite) ∧ (∀ n > 1, ∃ k, Nat.gcd n (a k) = 1) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/276.lean | erdos_276 | /--
Is there an infinite Lucas sequence $a_0, a_1, \ldots$ where $a_{n+2} = a_{n+1} + a_n$ for
$n \ge 0$ such that all $a_k$ are composite, and yet no integer has a common factor with every
term of the sequence?
-/ |
erdos_28 (A : Set ℕ) (h : (A + A)ᶜ.Finite) : limsup (fun (n : ℕ) => (sumRep A n : ℕ∞)) atTop = (⊤ : ℕ∞) := by sorry -- TODO(firsching): add the theorems/conjectures for the comments on the page | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/28.lean | erdos_28 | /--
If $A ⊆ \mathbb{N}$ is such that $A + A$ contains all but finitely many integers then
$\limsup 1_A ∗ 1_A(n) = \infty$.
-/ |
Condition (p : ℤ[X]) : Prop := p.leadingCoeff > 0 → ¬ (∃ d ≥ 2, ∀ n ≥ 1, d ∣ p.eval n) → ∀ᶠ m in atTop, ∃ k ≥ 1, ∃ n : Fin (k + 1) → ℤ, 0 = n 0 ∧ StrictMono n ∧ 1 = ∑ i ∈ Finset.Icc 1 (Fin.last k), (1 : ℚ) / (n i) ∧ m = ∑ i ∈ Finset.Icc 1 (Fin.last k), p.eval (n i) /-- Let $p\colon \mathbb{Z} \rightarrow \mathbb{Z}$ be... | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/283.lean | Condition | /--
Given a polynomial `p`, the predicate that if the leading coefficient is positive and
there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$, then for all sufficiently large $m$,
there exist integers $1≤n_1<\dots < n_k$ such that $$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$
and $$m=p(n_1)+\cdots+p(n_k)$$?
-/ |
erdos_283 : answer(sorry) ↔ ∀ p : ℤ[X], Condition p := by sorry /-- Graham [Gr63] has proved this when $p(x)=x$. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/283.lean | erdos_283 | /--
Let $p\colon \mathbb{Z} \rightarrow \mathbb{Z}$ be a polynomial whose leading coefficient is
positive and such that there exists no $d≥2$ with $d ∣ p(n)$ for all $n≥1$. Is it true that,
for all sufficiently large $m$, there exist integers $1≤n_1<\dots < n_k$ such that
$$1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$$
and
$$... |
erdos_283.variants.graham : Condition X := by sorry -- TODO(firsching): formalize the rest of the additional material | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/283.lean | erdos_283.variants.graham | /--
Graham [Gr63] has proved this when $p(x)=x$.
-/ |
erdos_285 : answer(True) ↔ ∀ᵉ (f : ℕ → ℕ) (S : Set ℕ) (hS : S = {k | ∃ (n : Fin k.succ → ℕ), StrictMono n ∧ 0 ∉ Set.range n ∧ 1 = ∑ i, (1 : ℝ) / n i }) (h : ∀ k ∈ S, IsLeast { n (Fin.last k) | (n : Fin k.succ → ℕ) (_ : StrictMono n) (_ : 0 ∉ Set.range n) (_ : 1 = ∑ i, (1 : ℝ) / n i) } (f k)), ∃ (o : ℕ → ℝ) (_ : o =o[at... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/285.lean | erdos_285 | /--
Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \dots < n_k$ with
$$
1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}.
$$
Is it true that
$$
f(k) = (1 + o(1)) \frac{e}{e - 1} k ?
$$
Proved by Martin [Ma00].
[Ma00] Martin, Greg, _Denser Egyptian fractions_. Acta Arith. (2000), 231-260.
-/ |
erdos_285.variants.lb (f : ℕ → ℕ) (S : Set ℕ) (hS : S = {k | ∃ (n : Fin k.succ → ℕ), StrictMono n ∧ 0 ∉ Set.range n ∧ 1 = ∑ i, (1 : ℝ) / n i }) (h : ∀ k ∈ S, IsLeast { n (Fin.last k) | (n : Fin k.succ → ℕ) (_ : StrictMono n) (_ : 0 ∉ Set.range n) (_ : 1 = ∑ i, (1 : ℝ) / n i) } (f k)) : ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/285.lean | erdos_285.variants.lb | /--
It is trivial that $f(k)\geq (1 + o(1)) \frac{e}{e - 1}k$.
-/ |
erdos_288 : answer(sorry) ↔ Set.Finite { I : Fin 2 → ℕ+ × ℕ+ | ∃ n : ℕ+, (∑ j : Fin 2, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } := by sorry /-- This is still open even if $|I_2| = 1$. -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/288.lean | erdos_288 | /--
Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that
$$
\sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2} \in \mathbb{N}?
$$
-/ |
erdos_288.variants.i2_card_eq_1 : answer(sorry) ↔ Set.Finite { (I, n₂) : (ℕ+ × ℕ+) × ℕ+ | ∃ n : ℕ+, ∑ n₁ ∈ Set.Icc I.1 I.2, (n₁⁻¹ : ℚ) + (n₂⁻¹ : ℚ) = n } := by sorry /-- It is perhaps true with two intervals replaced by any $k$ intervals. -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/288.lean | erdos_288.variants.i2_card_eq_1 | /--
This is still open even if $|I_2| = 1$.
-/ |
erdos_288.variants.k_intervals : answer(sorry) ↔ ∀ k, Set.Finite { I : Fin k → ℕ+ × ℕ+ | ∃ n : ℕ+, (∑ j : Fin k, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } := by sorry /-- Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/288.lean | erdos_288.variants.k_intervals | /--
It is perhaps true with two intervals replaced by any $k$ intervals.
-/ |
erdos_288.variants.exists_k_gt_2 : answer(sorry) ↔ ∃ k > 2, Set.Finite { I : Fin k → ℕ+ × ℕ+ | ∃ n : ℕ+, (∑ j : Fin k, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n } := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/288.lean | erdos_288.variants.exists_k_gt_2 | /--
Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?
-/ |
erdos_289 : answer(sorry) ↔ (∀ᶠ k : ℕ in atTop, ∃ I : Fin k → Finset ℕ, (∀ i, 2 ≤ #(I i) ∧ ∃ a b, 0 < a ∧ I i = Finset.Icc a b) ∧ ∑ i, ∑ n ∈ I i, (n⁻¹ : ℚ) = 1) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/289.lean | erdos_289 | /-- Is it true that, for all sufficiently large $k$, there exists finite intervals
$I_1, \dotsc, I_k \subset \mathbb{N}$ with $|I_i| \geq 2$ for $1 \leq i \leq k$ such that
$$
1 = \sum_{i=1}^k \sum_{n \in I_i} \frac{1}{n}.
$$
-/ |
exists_k (N : ℕ) : ∃ (k : ℕ) (n : Fin k.succ → ℕ), (∀ i, N ≤ n i) ∧ StrictMono n ∧ ∑ i, (1 / n i : ℝ) = 1 := by sorry /-- Let $k(N)$ denote the smallest $k$ such that there exists $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$. -/ | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/295.lean | exists_k | /--
Helper lemma: for each $N$, there exists $k$ and $n_1 < ... < n_k$ such that
$N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$.
-/ |
k (N : ℕ) : ℕ := Nat.find (exists_k N) /-- Let $k(N)$ denote the smallest $k$ such that there exists $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$ Is it true that $\lim_{N \to \infty} k(N) - (e - 1)N = \infty$? -/ @[category research open, AMS 5 11] | abbrev | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/295.lean | k | /--
Let $k(N)$ denote the smallest $k$ such that there exists
$N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$.
-/ |
erdos_295 : answer(sorry) ↔ Filter.atTop.Tendsto (fun N => k N - (rexp 1 - 1)*N) Filter.atTop := by sorry /-- Erdős and Straus have proved the existence of some constant $c>0$ such that $-c < k(N)-(e-1)N \ll \frac N {\log N}$ -/ @[category research solved, AMS 5 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/295.lean | erdos_295 | /--
Let $k(N)$ denote the smallest $k$ such that there exists
$N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$
Is it true that $\lim_{N \to \infty} k(N) - (e - 1)N = \infty$?
-/ |
erdos_295.variants.erdos_straus : ∃ᵉ (C > 0) (O > 0), ∀ᶠ (N : ℕ) in Filter.atTop, (k N - (rexp 1 - 1)*N) ∈ Set.Ioc (-C) (O * N / (N : ℝ).log):= by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/295.lean | erdos_295.variants.erdos_straus | /--
Erdős and Straus have proved the existence of some constant $c>0$
such that $-c < k(N)-(e-1)N \ll \frac N {\log N}$
-/ |
erdos_298 : answer(True) ↔ (∀ (A : Set ℕ), 0 ∉ A → A.HasPosDensity → ∃ (S : Finset ℕ), S.toSet ⊆ A ∧ ∑ n ∈ S, (1 / n : ℚ) = 1) := by sorry /-- In [Bl21] it is proved under the weaker assumption that `A` only has positive upper density. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/298.lean | erdos_298 | /-- Does every set $A \subseteq \mathbb{N}$ of positive density contain some finite $S \subset A$ such that
$\sum_{n \in S} \frac{1}{n} = 1$?
The answer is yes, proved by Bloom [Bl21].
[Bl21] Bloom, T. F., _On a density conjecture about unit fractions_. arXiv:2112.03726 (2021).
Note: The solution to this problem has... |
erdos_298.variants.upper_density : answer(True) ↔ (∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ (S : Finset ℕ), S.toSet ⊆ A ∧ ∑ n ∈ S, (1 / n : ℚ) = 1) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/298.lean | erdos_298.variants.upper_density | /--
In [Bl21] it is proved under the weaker assumption that `A` only has positive upper density.
-/ |
erdos_299 : answer(False) ↔ (∃ (a : ℕ → ℕ), StrictMono a ∧ (∀ n, 0 < a n) ∧ (fun n ↦ (a (n + 1) : ℝ) - a n) =O[atTop] (1 : ℕ → ℝ) ∧ ∀ S : Finset ℕ, ∑ i ∈ S, (1 : ℝ) / a i ≠ 1) := by sorry /-- The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive densit... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/299.lean | erdos_299 | /--
Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite
sum of $\frac{1}{a_i}$ is equal to 1?
-/ |
erdos_299.variants.density : ∀ (A : Set ℕ), 0 ∉ A → 0 < A.upperDensity → ∃ S : Finset ℕ, S.toSet ⊆ A ∧ ∑ n ∈ S, (1 : ℝ) / n = 1 := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/299.lean | erdos_299.variants.density | /--
The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$
with sets of positive density, as follows from [Bl21].
The statement is as follows:
If $A \subset \mathbb{N}$ has positive upper density (and hence certainly if $A$ has positive
density) then there is a finite $S \s... |
erdos_3 : answer(sorry) ↔ ∀ A : Set ℕ, (¬ Summable fun a : A ↦ 1 / (a : ℝ)) → ∃ᶠ (k : ℕ) in Filter.atTop, ∃ S ⊆ A, S.IsAPOfLength k := by sorry --TODO(firsching): add the various known bounds as variants. | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/3.lean | erdos_3 | /--
If $A \subset \mathbb{N} has $\sum_{n \in A}\frac 1 n = \infty$, then must $A$ contain arbitrarily
long arithmetic progressions?
-/ |
h (N : ℕ) : ℕ := Finset.maxSidonSubsetCard (Finset.Icc 1 N) open Filter /-- Is it true that, for every $\varepsilon > 0$, $h(N) = \sqrt N + O_{\varespilon}(N^\varespilon) -/ @[category research open, AMS 11] | abbrev | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/30.lean | h | /--
Let $h(N)$ be the maximum size of a Sidon set in $\{1, \dots, N\}$.
-/ |
erdos_30 : answer(sorry) ↔ ∀ᵉ (ε > 0), (fun N => h N - (N : Real).sqrt) =O[atTop] fun N => (N : ℝ)^(ε : ℝ) := by sorry -- TODO(firsching): add the various known bounds as variants. | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/30.lean | erdos_30 | /--
Is it true that, for every $\varepsilon > 0$, $h(N) = \sqrt N + O_{\varespilon}(N^\varespilon)
-/ |
erdos_303 : answer(True) ↔ --For any finite colouring of the integers ∀ (𝓒 : ℤ → ℤ), (Set.range 𝓒).Finite → --There exists integers `a, b, c` ∃ (a b c : ℤ), --that are non-zero and distinct. [a, b, c, 0].Nodup ∧ --`a, b, c` satisfy the equation (1/a : ℝ) = 1/b + 1/c ∧ --`a, b, c` have the same color (𝓒 '' {a, b, c})... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/303.lean | erdos_303 | /-- Is it true that in any finite colouring of the integers there exists a monochromatic solution
to $\frac 1 a = \frac 1 b + \frac 1 c$ with distinct $a, b, c$?
This is true, as proved by Brown and Rödl [BrRo91].
[BrRo91] Brown, Tom C. and Rödl, Voijtech,
_Monochromatic solutions to equations with unit fractions_.
B... |
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] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | unitFractionExpressible | /--
The set of `k` for which `a / b` can be expressed as a sum of `k` distinct unit fractions.
-/ |
zero_mem_unitFractionExpressible_iff {a b : ℕ} : 0 ∈ unitFractionExpressible a b ↔ a = 0 ∨ b = 0 := by simp_all [unitFractionExpressible] @[category API, AMS 11] | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | zero_mem_unitFractionExpressible_iff | /--
The set of `k` for which `a / b` can be expressed as a sum of `k` distinct unit fractions.
-/ |
unitFractionExpressible_of_zero {a b : ℕ} (h : a = 0 ∨ b = 0) : unitFractionExpressible a b = {0} := by simp only [Set.eq_singleton_iff_unique_mem, zero_mem_unitFractionExpressible_iff, *] have : (a / b : ℚ) = 0 := by simpa simp only [unitFractionExpressible, gt_iff_lt, Set.mem_setOf_eq, forall_exists_index, and_imp, t... | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | unitFractionExpressible_of_zero | /--
The set of `k` for which `a / b` can be expressed as a sum of `k` distinct unit fractions.
-/ |
unitFractionExpressible_zero_left {b : ℕ} : unitFractionExpressible 0 b = {0} := unitFractionExpressible_of_zero (by simp) @[category API, AMS 11] | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | unitFractionExpressible_zero_left | null |
unitFractionExpressible_zero_right {a : ℕ} : unitFractionExpressible a 0 = {0} := unitFractionExpressible_of_zero (by simp) @[category API, AMS 11] | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | unitFractionExpressible_zero_right | null |
zero_notMem_unitFractionExpressible {a b : ℕ} : 0 ∉ unitFractionExpressible a b ↔ a ≠ 0 ∧ b ≠ 0 := by simp_all [unitFractionExpressible] @[category API, AMS 11] | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | zero_notMem_unitFractionExpressible | null |
eq_inv_of_one_mem_unitFractionExpressible {a b : ℕ} (h : 1 ∈ unitFractionExpressible a b) : ∃ m : ℕ, 1 < m ∧ (a / b : ℚ) = (m : ℚ)⁻¹ := by simp only [unitFractionExpressible, gt_iff_lt, Set.mem_setOf_eq, Finset.card_eq_one] at h obtain ⟨_, ⟨m, rfl⟩, h₁, h₂⟩ := h simp only [Finset.mem_singleton, forall_eq, Finset.sum_si... | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | eq_inv_of_one_mem_unitFractionExpressible | null |
dvd_of_one_mem_unitFractionExpressible {a b : ℕ} (h : 1 ∈ unitFractionExpressible a b) : a ∣ b := by obtain ⟨m, hm₁, hm⟩ := eq_inv_of_one_mem_unitFractionExpressible h have : b ≠ 0 := by rintro rfl simp [eq_comm] at hm omega use m field_simp at hm exact mod_cast hm.symm /-- Let $$N(a, b)$$, denoted here by `smallestCol... | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | dvd_of_one_mem_unitFractionExpressible | null |
smallestCollection (a b : ℕ) : ℕ := sInf (unitFractionExpressible a b) -- in fact `(unitFractionExpressible a b).Nonempty` should always be true, but we do not prove it -- for now @[category API, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | smallestCollection | /-- Let $$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}$$ -/ |
smallestCollection_pos {a b : ℕ} (ha : a ≠ 0) (hb : b ≠ 0) (h : (unitFractionExpressible a b).Nonempty) : 0 < smallestCollection a b := by suffices smallestCollection a b ≠ 0 by omega intro h' have : 0 ∈ unitFractionExpressible a b := h' ▸ Nat.sInf_mem h simp_all @[category API, AMS 11] | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | smallestCollection_pos | /-- Let $$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}$$ -/ |
smallestCollection_left_one (b : ℕ) (hb : 1 < b) : smallestCollection 1 b = 1 := by have : 1 ∈ unitFractionExpressible 1 b := ⟨{b}, by simpa⟩ have : smallestCollection 1 b ≤ 1 := Nat.sInf_le this have : 0 ∉ unitFractionExpressible 1 b := by simp; omega have : smallestCollection 1 b ≠ 0 := ne_of_mem_of_not_mem (Nat.sInf... | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | smallestCollection_left_one | null |
eq_one_of_smallestCollection_eq_one {a b : ℕ} (h : smallestCollection a b = 1) : ∃ m : ℕ, 1 < m ∧ (a / b : ℚ) = (m : ℚ)⁻¹ := by have : 1 ∈ unitFractionExpressible a b := h ▸ Nat.sInf_mem (Nat.nonempty_of_sInf_eq_succ h) apply eq_inv_of_one_mem_unitFractionExpressible this @[category API, AMS 11] | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | eq_one_of_smallestCollection_eq_one | null |
dvd_of_smallestCollection_eq_one {a b : ℕ} (h : smallestCollection a b = 1) : a ∣ b := by have : 1 ∈ unitFractionExpressible a b := h ▸ Nat.sInf_mem (Nat.nonempty_of_sInf_eq_succ h) apply dvd_of_one_mem_unitFractionExpressible this @[category test, AMS 11] | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | dvd_of_smallestCollection_eq_one | null |
smallestCollection_two_fifteen : smallestCollection 2 15 = 2 := by have h : 2 ∈ unitFractionExpressible 2 15 := by use {10, 30} norm_num [Finset.card_insert_of_notMem, Finset.card_singleton] have : smallestCollection 2 15 ≤ 2 := Nat.sInf_le h have : 0 < smallestCollection 2 15 := smallestCollection_pos (by simp) (by si... | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | smallestCollection_two_fifteen | null |
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$$. [Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg\'{E}sz sz\'{A}m\'{u} megold\'{A}sairól. Mat. Lapok (1950), 192-2... | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | smallestCollectionTo | /-- Write $$N(b) = max_{1 \leq a < b} N(a, b)$$. -/ |
erdos_304.variants.upper_1950 : (fun b => (smallestCollectionTo b : ℝ)) =O[atTop] (fun b => Real.log b / Real.log (Real.log b)) := by sorry /-- In 1950, Erdős [Er50c] proved the lower bound $$\log \log b \ll N(b)$$. [Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg\'{E}sz sz\'{A}m\'{u}... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | erdos_304.variants.upper_1950 | /--
In 1950, Erdős [Er50c] proved the upper bound $$N(b) \ll \log b / \log \log b$$.
[Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg\'{E}sz sz\'{A}m\'{u} megold\'{A}sairól. Mat. Lapok (1950), 192-210.
-/ |
erdos_304.variants.lower_1950 : (fun b : ℕ => Real.log (Real.log b)) =O[atTop] (fun b => (smallestCollectionTo b : ℝ)) := by sorry /-- In 1985 Vose [Vo85] proved the upper bound $$N(b) \ll \sqrt{\log b}$$. [Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24. -/ @[category research solved,... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | erdos_304.variants.lower_1950 | /--
In 1950, Erdős [Er50c] proved the lower bound $$\log \log b \ll N(b)$$.
[Er50c] Erdős, P., Az ${1}/{x_1} + {1}/{x_2} + \ldots + {1}/{x_n} =A/B$ egyenlet eg\'{E}sz sz\'{A}m\'{u} megold\'{A}sairól. Mat. Lapok (1950), 192-210.
-/ |
erdos_304.variants.upper_1985 : (fun b => (smallestCollectionTo b : ℝ)) =O[atTop] (fun b => Real.sqrt (Real.log b)) := sorry /-- Is it true that $$N(b) \ll \log \log b$$? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | erdos_304.variants.upper_1985 | /--
In 1985 Vose [Vo85] proved the upper bound $$N(b) \ll \sqrt{\log b}$$.
[Vo85] Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.
-/ |
upper_bound : answer(sorry) ↔ (fun b : ℕ => (smallestCollectionTo b : ℝ)) =O[atTop] (fun b : ℕ => Real.log (Real.log b)) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/304.lean | upper_bound | /--
Is it true that $$N(b) \ll \log \log b$$?
-/ |
erdos_306 : answer(sorry) ↔ ∀ (q : ℚ), 0 < q → Squarefree q.den → ∃ k : ℕ, ∃ (n : Fin (k + 1) → ℕ), n 0 = 1 ∧ StrictMono n ∧ (∀ i ∈ Finset.Icc 1 (Fin.last k), ω (n i) = 2 ∧ Ω (n i) = 2) ∧ q = ∑ i ∈ Finset.Icc 1 (Fin.last k), (1 : ℚ) / (n i) := by sorry /-- Every positive integer can be expressed as an Egyptian fraction... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/306.lean | erdos_306 | /--
Let $\frac a b\in \mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1 < n_1 < \dots < n_k$,
each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$?
-/ |
erdos_306.variant.integer_three_primes (m : ℕ) (h : 0 < m) : ∃ k > (0 : ℕ), ∃ (n : Fin (k + 1) → ℕ), n 0 = 1 ∧ ∀ i, (hik : i < k) → n ⟨i, by omega⟩ < n ⟨(i + 1), by omega⟩ ∧ (∀ i ∈ Finset.Icc 1 (Fin.last k), ω (n i) = 3 ∧ Ω (n i) = 3) ∧ m = ∑ i ∈ Finset.Icc 1 (Fin.last k), (1 : ℚ) / (n i) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/306.lean | erdos_306.variant.integer_three_primes | /--
Every positive integer can be expressed as an Egyptian fraction where each denominator is the
product of three distinct primes.
-/ |
erdos_307 : answer(sorry) ↔ ∃ P Q : Finset ℕ, (∀ p ∈ P, p.Prime) ∧ (∀ q ∈ Q, q.Prime) ∧ (1 = (∑ p ∈ P, (p : ℚ)⁻¹) * (∑ q ∈ Q, (q : ℚ)⁻¹)) := by sorry /-- Instead of asking for sets of primes, ask only that all primes in the sets be relatively coprime. -/ @[category research solved, AMS 5 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/307.lean | erdos_307 | /--
Are there two finite set of primes $P$ and $Q$ such that
$$
1 = \left( \sum_{p \in P} \frac{1}{p} \right) \left( \sum_{q \in Q} \frac{1}{q} \right)
$$
?
Asked by Barbeau [Ba76].
[Ba76] Barbeau, E. J., _Computer challenge corner: Problem 477: A brute force program._
-/ |
erdos_307_coprime : answer(sorry) ↔ ∃ P Q : Finset ℕ, 1 < #P ∧ 1 < #Q ∧ Set.Pairwise P Nat.Coprime ∧ Set.Pairwise Q Nat.Coprime ∧ (1 = (∑ p ∈ P, (p : ℚ)⁻¹) * (∑ q ∈ Q, (q : ℚ)⁻¹)) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/307.lean | erdos_307_coprime | /--
Instead of asking for sets of primes, ask only that all primes in the sets be relatively coprime.
-/ |
erdos_312 : answer(sorry) ↔ ∃ (c : ℝ), 0 < c ∧ ∀ (K : ℝ), 1 < K → ∃ (N₀ : ℕ), ∀ (n : ℕ) (a : Fin n → ℕ), (n ≥ N₀ ∧ (∑ i : Fin n, (a i : ℝ)⁻¹) > K) → ∃ (S : Finset (Fin n)), 1 - Real.exp (-(c * K)) < (∑ i ∈ S, (a i : ℝ)⁻¹) ∧ ∑ i ∈ S, (a i : ℝ)⁻¹ ≤ 1 := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/312.lean | erdos_312 | /--
Does there exist a constant `c > 0` such that, for any `K > 1`, whenever `A` is a sufficiently large
finite multiset of integers with $\sum_{n \in A} 1/n > K$ there exists some $S \subseteq A$ such that
$1 - \exp(-(c*K)) < \sum_{n \in S} 1/n \le 1$?
-/ |
erdos_313_solutions : Set (ℕ × Finset ℕ) := {(m, P) | 2 ≤ m ∧ P.Nonempty ∧ (∀ p ∈ P, p.Prime) ∧ ∑ p ∈ P, (1 : ℚ) / p = 1 - 1 / m} /-- Are there infinitely many pairs `(m, P)` where `m ≥ 2` is an integer and `P` is a set of distinct primes such that the following equation holds: $\sum_{p \in P} \frac{1}{p} = 1 - \frac{1... | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/313.lean | erdos_313_solutions | /--
This set contains all solutions `(m, P)` to the Erdős problem 313.
A solution is a pair where `m` is an integer `≥ 2` and `P` is a non-empty, finite set of
distinct prime numbers, such that the sum of the reciprocals of the primes in `P` equals `1 - 1/m`.
-/ |
erdos_313_conjecture : answer(sorry) ↔ erdos_313_solutions.Infinite := by sorry @[category test, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/313.lean | erdos_313_conjecture | /--
Are there infinitely many pairs `(m, P)` where `m ≥ 2` is an integer
and `P` is a set of distinct primes such that the following equation holds:
$\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}$?
-/ |
erdos_313_solution_6_2_3 : (6, {2, 3}) ∈ erdos_313_solutions := by norm_num [erdos_313_solutions] @[category test, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/313.lean | erdos_313_solution_6_2_3 | /--
Are there infinitely many pairs `(m, P)` where `m ≥ 2` is an integer
and `P` is a set of distinct primes such that the following equation holds:
$\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}$?
-/ |
erdos_313_solution_42_2_3_7 : (42, {2, 3, 7}) ∈ erdos_313_solutions := by norm_num [erdos_313_solutions] /-- An integer `n` is a **primary pseudoperfect number** if it is the denominator `m` in a solution `(m, P)` to the Erdős 313 problem. -/ | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/313.lean | erdos_313_solution_42_2_3_7 | /--
Are there infinitely many pairs `(m, P)` where `m ≥ 2` is an integer
and `P` is a set of distinct primes such that the following equation holds:
$\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}$?
-/ |
IsPrimaryPseudoperfect (n : ℕ) : Prop := ∃ P, (n, P) ∈ erdos_313_solutions /-- It is conjectured that the set of primary pseudoperfect numbers is infinite. -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/313.lean | IsPrimaryPseudoperfect | /--
An integer `n` is a **primary pseudoperfect number** if it is the denominator `m` in a
solution `(m, P)` to the Erdős 313 problem.
-/ |
erdos_313.variant.primary_pseudoperfect_are_infinite : Set.Infinite {n | IsPrimaryPseudoperfect n} := by sorry /-- There are at least 8 primary pseudoperfect numbers. -/ @[category undergraduate, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/313.lean | erdos_313.variant.primary_pseudoperfect_are_infinite | /--
It is conjectured that the set of primary pseudoperfect numbers is infinite.
-/ |
exists_at_least_eight_primary_pseudoperfect : 8 ≤ (Set.encard {n | IsPrimaryPseudoperfect n}) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/313.lean | exists_at_least_eight_primary_pseudoperfect | /--
There are at least 8 primary pseudoperfect numbers.
-/ |
erdos_316 : answer(False) ↔ ∀ A : Finset ℕ, 0 ∉ A → 1 ∉ A → ∑ n ∈ A, (1 / n : ℚ) < 2 → ∃ (A₁ A₂ : Finset ℕ), Disjoint A₁ A₂ ∧ A = A₁ ∪ A₂ ∧ ∑ n ∈ A₁, (1 / n : ℚ) < 1 ∧ ∑ n ∈ A₂, (1 / n : ℚ) < 1 := by simp only [one_div, false_iff, not_forall, not_exists, not_and, not_lt] let A : Finset ℕ := {2, 3, 4, 5, 6, 7, 10, 11, 1... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/316.lean | erdos_316 | /-- Is it true that if $A \subseteq \mathbb{N}\setminus\{1\}$ is a finite set with
$\sum_{n \in A} \frac{1}{n} < 2$ then there is a partition $A=A_1 \sqcup A_2$
such that $\sum_{n \in A_i} \frac{1}{n} < 1$ for $i=1,2$?
This is not true in general, as shown by Sándor [Sa97].
[Sa97] S\'{A}ndor, Csaba, _On a problem of ... |
erdos_316.variants.multiset : ∃ A : Multiset ℕ, 0 ∉ A ∧ 1 ∉ A ∧ (A.map ((1 : ℚ) / ·)).sum < 2 ∧ ∀ (A₁ A₂ : Multiset ℕ), A = A₁ + A₂ → 1 ≤ (A₁.map ((1 : ℚ) / ·)).sum ∨ 1 ≤ (A₂.map ((1 : ℚ) / ·)).sum := by let A : Multiset ℕ := {2, 3, 3, 5, 5, 5, 5} refine ⟨A, by decide, by decide, by decide +kernel, ?_⟩ suffices h : ∀ B... | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/316.lean | erdos_316.variants.multiset | /-- It is not true if `A` is a multiset (easier) -/ |
erdos_316.variants.generalized (n : ℕ) (hn : 2 ≤ n) : ∃ A : Finset ℕ, A.Nonempty ∧ 0 ∉ A ∧ 1 ∉ A ∧ ∑ k ∈ A, (1 / k : ℚ) < n ∧ ∀ P : Finpartition A, P.parts.card = n → ∃ p ∈ P.parts, 1 ≤ ∑ n ∈ p, (1 / n : ℚ) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/316.lean | erdos_316.variants.generalized | /-- More generally, Sándor shows that for any $n \ge 2$ there exists a finite set
$A \subseteq \mathbb{N}\setminus\{1\}$ with $\sum_{k \in A} \frac{1}{k} < n$, and no
partition into $n$ parts each of which has $\sum_{n \in A_i} \frac{1}{n} < 1$. -/ |
erdos_317 : answer(sorry) ↔ ∃ c > 0, ∀ n ≥ 1, ∃ δ : Fin n → ℚ, Set.range δ ⊆ {-1, 0, 1} ∧ letI lhs : ℝ := |∑ k, (δ k) / (k + 1)| 0 < lhs ∧ lhs < c / 2^n := by sorry /-- Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$, \[\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/317.lean | erdos_317 | /--
Is there some constant $c>0$ such that for every $n\geq 1$ there exists some $\delta_k\in \{-1,0,1\}$ for $1\leq k\leq n$ with
\[0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?\]
-/ |
erdos_317.variants.claim2 : answer(sorry) ↔ ∀ᶠ n in atTop, ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} → letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))| lhs ≠ 0 → lhs > 1 / (Icc 1 n).lcm id := by sorry /-- Inequality in `erdos_317.variants.claim2` is obvious, the problem is strict inequality. -/ @[category undergraduate, AMS ... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/317.lean | erdos_317.variants.claim2 | /--
Is it true that for sufficiently large $n$, for any $\delta_k\in \{-1,0,1\}$,
\[\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}\]
whenever the left-hand side is not zero?
-/ |
claim2_inequality : ∀ᶠ n in atTop, ∀ δ : (Fin n) → ℚ, δ '' Set.univ ⊆ {-1,0,1} → letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))| lhs ≠ 0 → lhs ≥ 1 / (Icc 1 n).lcm id := by sorry /-- `erdos_317.variants.claim2` fails for small $n$, for example \[\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.\] -/ @[category graduate, AMS 1... | lemma | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/317.lean | claim2_inequality | /--
Inequality in `erdos_317.variants.claim2` is obvious, the problem is strict inequality.
-/ |
erdos_317.variants.counterexample : ¬ (∀ δ : (Fin 4) → ℚ, δ '' Set.univ ⊆ {-1,0,1} → letI lhs := |∑ k, ((δ k : ℚ) / (k + 1))| lhs ≠ 0 → lhs > (1 : ℚ) / ((Icc 1 4).lcm id : ℕ)) := by push_neg use ![0, 1, -1, -1] norm_num [Finset.sum] refine ⟨by grind, le_of_eq ?_⟩ exact (abs_of_nonneg (by norm_num)).trans (one_div _) | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/317.lean | erdos_317.variants.counterexample | /--
`erdos_317.variants.claim2` fails for small $n$, for example
\[\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.\]
-/ |
erdos_319 (N : ℕ) : IsGreatest { #A | (A) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } answer(sorry) := by sorry -- Formalisation note: it's possible that solution to `erdos_319` needs to be -- expressed asymptotically. To handle this we... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/319.lean | erdos_319 | /-- What is the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function
$\delta : A \to \{-1, 1\}$ such that
$$
\sum_{n\in A} \frac{\delta n}{n} = 0
$$
and
$$
\sum_{n\in A'}\frac{\delta n}{n} \neq 0
$$
for all non-empty $A'\subsetneq A$.-/ |
erdos_319.variants.isTheta (N : ℕ) (c : ℕ → ℝ) (h : ∀ N, IsGreatest { (#A : ℝ) | (A) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } (c N)) : c =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by sorry /-- Let $c(N)$ be the size of the largest $A\subse... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/319.lean | erdos_319.variants.isTheta | /-- Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function
$\delta : A \to \{-1, 1\}$ such that
$$
\sum_{n\in A} \frac{\delta n}{n} = 0
$$
and
$$
\sum_{n\in A'}\frac{\delta n}{n} \neq 0
$$
for all non-empty $A'\subsetneq A$. What is $\Theta(c(N))$?-/ |
erdos_319.variants.isBigO (N : ℕ) (c : ℕ → ℝ) (h : ∀ N, IsGreatest { (#A : ℝ) | (A) (_ : A ⊆ Finset.Icc 1 N) (_ : ∃ δ : ℕ → ℤˣ, ∑ n ∈ A, (δ n : ℚ) / n = 0 ∧ ∀ A' ⊂ A, A'.Nonempty → ∑ n ∈ A', (δ n : ℚ) / n ≠ 0) } (c N)) : c =O[atTop] (answer(sorry) : ℕ → ℝ) := by sorry /-- Let $c(N)$ be the size of the largest $A\subset... | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/319.lean | erdos_319.variants.isBigO | /-- Let $c(N)$ be the size of the largest $A\subseteq\{1, \dots, N\}$ such that there is a function
$\delta : A \to \{-1, 1\}$ such that
$$
\sum_{n\in A} \frac{\delta n}{n} = 0
$$
and
$$
\sum_{n\in A'}\frac{\delta n}{n} \neq 0
$$
for all non-empty $A'\subsetneq A$. Find the simplest $g(N)$ such that $c(N) = O(g(N))... |
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