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erdos_12.variants.erdos_sarkozy (f : ℕ → ℕ) (hf : atTop.Tendsto f atTop) : ∃ A, IsGood A ∧ {N : ℕ | (N : ℝ) / f N < (A.interIcc 1 N).ncard}.Infinite := by sorry /-- An example of an $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$ and such that \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N > 0\] is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/12.lean
|
erdos_12.variants.erdos_sarkozy
|
/--
Given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with the property
that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$, such that there are
infinitely many $N$ such that \[\lvert A\cap\{1,\ldots,N\}\rvert > \frac{N}{f(N)}.
-/
|
erdos_12.variants.example (A : Set ℕ) (hA : A = {p ^ 2 | (p : ℕ) (_ : p.Prime) (_ : p ≡ 3 [MOD 4])}) : IsGood A ∧ 0 < atTop.liminf (fun (N : ℕ) ↦ (A.interIcc 1 N).ncard * (N : ℝ).log / √N) := by sorry /-- Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then \[\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}\] -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/12.lean
|
erdos_12.variants.example
|
/--
An example of an $A$ with the property that there are no distinct $a,b,c \in A$ such that
$a \mid (b+c)$ and $b,c > a$ and such that
\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N > 0\]
is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime.
-/
|
erdos_12.variants.schoen (A : Set ℕ) (hA : IsGood A) (hA' : A.Pairwise Nat.Coprime) : (fun N ↦ ((A.interIcc 1 N).ncard : ℝ)) =O[atTop] (fun N ↦ (N : ℝ) ^ (2 / 3 : ℝ)) := by sorry /-- Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then \[\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}/\log N\] -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/12.lean
|
erdos_12.variants.schoen
|
/--
Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such
that $a \mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then
\[\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}\]
-/
|
erdos_12.variants.baier (A : Set ℕ) (hA : IsGood A) (hA' : A.Pairwise Nat.Coprime) : (fun N ↦ ((A.interIcc 1 N).ncard : ℝ)) =O[atTop] (fun N ↦ (N : ℝ) ^ (2 / 3 : ℝ) / (N : ℝ).log) := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/12.lean
|
erdos_12.variants.baier
|
/--
Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such
that $a \mid (b+c)$ and $b,c > a$. If all elements in $A$ are pairwise coprime then
\[\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}/\log N\]
-/
|
IsDComplete (A : Set ℕ) : Prop := ∀ᶠ n in atTop, ∃ s : Finset ℕ, (s : Set ℕ) ⊆ A ∧ -- The summands come from A IsAntichain (· ∣ ·) (s : Set ℕ) ∧ -- No summand divides another s.sum id = n -- They sum to n /-- Characterizes a "snug" finite set of natural numbers: all elements are within a multiplicative factor $(1 + ε)$ of the minimum. Specifically, for a finite set $A$ and $ε > 0$, all $a ∈ A$ satisfy $a < (1 + ε) · min(A)$. -/
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/123.lean
|
IsDComplete
|
/--
A set `A` of natural numbers is **d-complete** if every sufficiently large integer
is the sum of distinct elements of `A` such that no element divides another.
Reference: [ErLe96] Erdős, P. and Lewin, M., _$d$-complete sequences of integers_. Math. Comp. (1996).
-/
|
IsSnug (ε : ℝ) (A : Finset ℕ) : Prop := ∃ hA : A.Nonempty, ∀ a ∈ A, a < (1 + ε) * A.min' hA /-- Predicate for pairwise coprimality of three integers. Requires all three input values to be pairwise coprime to each other. -/
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/123.lean
|
IsSnug
|
/--
Characterizes a "snug" finite set of natural numbers:
all elements are within a multiplicative factor $(1 + ε)$ of the minimum.
Specifically, for a finite set $A$ and $ε > 0$, all $a ∈ A$ satisfy $a < (1 + ε) · min(A)$.
-/
|
PairwiseCoprime (a b c : ℕ) : Prop := Pairwise (Nat.Coprime.onFun ![a, b, c]) /-- **Erdős Problem #123** Let $a, b, c$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^k b^l c^m$ ($k, l, m ≥ 0$), none of which divide any other? Equivalently: is the set $\{a^k b^l c^m : k, l, m \geq 0\}$ d-complete? Note: For this not to reduce to the two-integer case, we need the integers to be greater than one and distinct. -/ @[category research open, AMS 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/123.lean
|
PairwiseCoprime
|
/--
Predicate for pairwise coprimality of three integers.
Requires all three input values to be pairwise coprime to each other.
-/
|
erdos_123 : answer(sorry) ↔ ∀ a > 1, ∀ b > 1, ∀ c > 1, PairwiseCoprime a b c → IsDComplete (↑(powers a) * ↑(powers b) * ↑(powers c)) := by sorry /-- Erdős and Lewin proved this conjecture when $a = 3$, $b = 5$, and $c = 7$. Reference: [ErLe96] Erdős, P. and Lewin, Mordechai, _$d$-complete sequences of integers_. Math. Comp. (1996), 837-840. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/123.lean
|
erdos_123
|
/--
**Erdős Problem #123**
Let $a, b, c$ be three integers which are pairwise coprime. Is every large integer
the sum of distinct integers of the form $a^k b^l c^m$ ($k, l, m ≥ 0$), none of which
divide any other?
Equivalently: is the set $\{a^k b^l c^m : k, l, m \geq 0\}$ d-complete?
Note: For this not to reduce to the two-integer case, we need the integers
to be greater than one and distinct.
-/
|
erdos_123.variants.erdos_lewin_3_5_7 : IsDComplete (↑(powers 3) * ↑(powers 5) * ↑(powers 7)) := by sorry /-- A simpler case: the set of numbers of the form $2^k 3^l$ ($k, l ≥ 0$) is d-complete. This was initially conjectured by Erdős in 1992, who called it a "nice and difficult" problem, but it was quickly proven by Jansen and others using a simple inductive argument: - If $n = 2m$ is even, apply the inductive hypothesis to $m$ and double all summands. - If $n$ is odd, let $3^k$ be the largest power of $3$ with $3^k ≤ n$, and apply the inductive hypothesis to $n - 3^k$ (which is even). Reference: [Er92b] Erdős, Paul, _Some of my favourite problems in various branches of combinatorics_. Matematiche (Catania) (1992), 231-240. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/123.lean
|
erdos_123.variants.erdos_lewin_3_5_7
|
/--
Erdős and Lewin proved this conjecture when $a = 3$, $b = 5$, and $c = 7$.
Reference: [ErLe96] Erdős, P. and Lewin, Mordechai,
_$d$-complete sequences of integers_. Math. Comp. (1996), 837-840.
-/
|
erdos_123.variants.powers_2_3 : IsDComplete (↑(powers 2) * ↑(powers 3)) := by sorry /-- A stronger conjecture for numbers of the form $2^k 3^l 5^j$. For any $ε > 0$, all large integers $n$ can be written as the sum of distinct integers $b_1 < ... < b_t$ of the form $2^k 3^l 5^j$ where $b_t < (1 + ϵ) b_1$. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/123.lean
|
erdos_123.variants.powers_2_3
| null |
erdos_123.variants.powers_2_3_5_snug : answer(sorry) ↔ ∀ ε > 0, ∀ᶠ n in atTop, ∃ A : Finset ℕ, (A : Set ℕ) ⊆ ↑(powers 2) * ↑(powers 3) * ↑(powers 5) ∧ IsSnug ε A ∧ ∑ x ∈ A, x = n := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/123.lean
|
erdos_123.variants.powers_2_3_5_snug
|
/--
A stronger conjecture for numbers of the form $2^k 3^l 5^j$.
For any $ε > 0$, all large integers $n$ can be written as the sum of distinct integers
$b_1 < ... < b_t$ of the form $2^k 3^l 5^j$ where $b_t < (1 + ϵ) b_1$.
-/
|
erdos_124 : answer(sorry) ↔ ∀ k, ∀ d : Fin k → ℕ, (∀ i, 3 ≤ d i) → StrictMono d → 1 ≤ ∑ i : Fin k, (1 : ℚ) / (d i - 1) → ∀ᶠ n in atTop, ∃ c : Fin k → ℕ, ∃ a : Fin k → ℕ, ∀ i, c i ∈ ({0, 1} : Finset ℕ) ∧ ∀ i, ((d i).digits (a i)).toFinset ⊆ {0, 1} ∧ n = ∑ i, c i * a i := by sorry -- TODO(firsching): formalize the other two claims from the additional material
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/124.lean
|
erdos_124
|
/--
Let $3\leq d_1 < d_2 < \cdots < d_k$ be integers such that
$$\sum_{1\leq i\leq k}\frac{1}{d_i-1}\geq 1.$$
Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$
where $c_i\in \{0, 1\}$ and $a_i$ has only the digits $0, 1$ when written in base $d_i$?
Conjectured by Burr, Erdős, Graham, and Li [BEGL96]
-/
|
erdos_125 : answer(sorry) ↔ ({ x : ℕ | (digits 3 x).toFinset ⊆ {0, 1} } + { x : ℕ | (digits 4 x).toFinset ⊆ {0, 1} }).HasPosDensity := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/125.lean
|
erdos_125
| null |
IsMaximalAddFactorsCard (f : ℕ → ℕ) : Prop := ∀ n, IsGreatest { m | ∀ (A : Finset ℕ), A.card = n → m ≤ (∏ ⟨a, b⟩ ∈ A.offDiag, (a + b)).primeFactors.card} (f n) /-- Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $|A| = n$ then $\prod_{a\neq b\in A}(a + b)$ has at least $f(n)$ distinct prime factors. Is it true that $\frac{f(n)}{\log n} \to\infty$? -/ @[category research open, AMS 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/126.lean
|
IsMaximalAddFactorsCard
| null |
erdos_126 : answer(sorry) ↔ ∀ (f : ℕ → ℕ), IsMaximalAddFactorsCard f → Tendsto (fun n => f n / Real.log n) atTop atTop := by sorry /-- Erdős and Turán proved [ErTu34] in their first joint paper that $$ \log n \ll f(n) \ll \frac{n}{\log n} $$ [ErTu34] Erdős, Paul and Turan, Paul, _On a Problem in the Elementary Theory of Numbers_. Amer. Math. Monthly (1934), 608-611. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/126.lean
|
erdos_126
|
/--
Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $|A| = n$ then
$\prod_{a\neq b\in A}(a + b)$ has at least $f(n)$ distinct prime factors.
Is it true that $\frac{f(n)}{\log n} \to\infty$?
-/
|
erdos_126.variants.IsBigO (f : ℕ → ℕ) (hf : IsMaximalAddFactorsCard f) : ((fun (n : ℕ) => Real.log n) =O[atTop] fun (n : ℕ) => (f n : ℝ)) ∧ (fun (n : ℕ) => (f n : ℝ)) =O[atTop] fun (n : ℕ) => n / Real.log n := by sorry /-- Erdős says that $f(n) = o(\frac{n}{\log n})$ has never been proved. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/126.lean
|
erdos_126.variants.IsBigO
|
/--
Erdős and Turán proved [ErTu34] in their first joint paper that
$$
\log n \ll f(n) \ll \frac{n}{\log n}
$$
[ErTu34] Erdős, Paul and Turan, Paul, _On a Problem in the Elementary Theory of Numbers_. Amer. Math. Monthly (1934), 608-611.
-/
|
erdos_126.variants.isLittleO (f : ℕ → ℕ) (hf : IsMaximalAddFactorsCard f) : (fun (n : ℕ) => (f n : ℝ)) =o[atTop] (fun (n : ℕ) => n / Real.log n) := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/126.lean
|
erdos_126.variants.isLittleO
|
/--
Erdős says that $f(n) = o(\frac{n}{\log n})$ has never been proved.
-/
|
erdos_128 : answer(sorry) ↔ (∀ V' : Set V, 2 * V'.ncard + 1 ≥ Fintype.card V → 50 * (G.induce V').edgeSet.ncard > Fintype.card V ^ 2) → ¬ G.CliqueFree 3 := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/128.lean
|
erdos_128
|
/--
Let G be a graph with n vertices such that every subgraph on ≥ $n/2$
vertices has more than $n^2/50$ edges. Must G contain a triangle?
-/
|
erdos_137 : answer(sorry) ↔ ∀ k ≥ 3, ∀ n, ¬ (∏ x ∈ Finset.Ioc n (n + k), x).Powerful := by sorry /-- Let $k\geq 2$. Erdős and Selfridge [ES75] proved that the product of any $k$ consecutive integers $N$ cannot be a perfect power. [ES75] P. Erdös, J. L. Selfridge, "The product of consecutive integers is never a power", Illinois J. Math. 19(2): 292-301, 1975 -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/137.lean
|
erdos_137
|
/--
Let $k\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is,
must there always exist a prime $p\mid N$ such that $p^2\nmid N$?
-/
|
erdos_137.variants.perfect_power (k : ℕ) (hk : k ≥ 2) (n : ℕ) (x l : ℕ) (hl : 2 ≤ l) : (∏ x ∈ Finset.Ioc n (n + k), x) ≠ x ^ l := by sorry /-- Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all positive integers $m$, there must be at least $k$ distinct primes $p$ such that $p\mid m(m+1)\cdots (m+n)$ and yet $p^2$ does not divide the right hand side. [Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45., -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/137.lean
|
erdos_137.variants.perfect_power
|
/--
Let $k\geq 2$. Erdős and Selfridge [ES75] proved that the product of any $k$ consecutive
integers $N$ cannot be a perfect power.
[ES75] P. Erdös, J. L. Selfridge, "The product of consecutive integers is never a power",
Illinois J. Math. 19(2): 292-301, 1975
-/
|
erdos_137.multiple_powerful_factors (k : ℕ) : ∀ᶠ n in Filter.atTop, ∀ (m : ℕ) (hm : 0 < m), letI N := ∏ x ∈ Finset.Ioc m (m + n), x ∃ P : Finset ℕ, P.card = k ∧ ∀ p ∈ P, p.Prime ∧ p ∣ N ∧ ¬ p ^ 2 ∣ N := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/137.lean
|
erdos_137.multiple_powerful_factors
|
/--
Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all
positive integers $m$, there must be at least $k$ distinct primes $p$ such that
$p\mid m(m+1)\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.
[Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45.,
-/
|
monoAP_guarantee_set (r k : ℕ) : Set ℕ := { N | ∀ coloring : Finset.Icc 1 N → Fin r, ContainsMonoAPofLength coloring k} /-- Asserts that for any number of colors `r` and any progression length `k`, there always exists some number `N` large enough to guarantee a monochromatic arithmetic progression. In other words, the set `monoAP_guarantee_set` is non-empty. This is the fundamental existence result that allows for the definition of the van der Waerden numbers. -/ @[category research solved, AMS 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
monoAP_guarantee_set
|
/--
The set of natural numbers that guarantee a monochromatic arithmetic progression.
A number `N` belongs to this set if, for a given number of colors `r` and an arithmetic
progression length `k`, any `r`-coloring of the integers `{1, ..., N}` must contain a
monochromatic arithmetic progression of length `k`.
-/
|
monoAP_guarantee_set_nonempty (r k) : (monoAP_guarantee_set r k).Nonempty := by sorry /-- The **van der Waerden number**, is the smallest integer `N` such that any `r`-coloring of `{1, ..., N}` is guaranteed to contain a monochromatic arithmetic progression of length `k`. It is defined as the infimum of the (non-empty) set of all such numbers `N`. -/
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
monoAP_guarantee_set_nonempty
|
/--
Asserts that for any number of colors `r` and any progression length `k`, there
always exists some number `N` large enough to guarantee a monochromatic arithmetic progression.
In other words, the set `monoAP_guarantee_set` is non-empty. This is the fundamental existence
result that allows for the definition of the van der Waerden numbers.
-/
|
monoAPNumber (r k : ℕ) : ℕ := sInf (monoAP_guarantee_set r k) /-- An abbreviation for the van der Waerden number for 2 colors, commonly written as `W(k)`. This represents the smallest integer `N` such that any 2-coloring of `{1, ..., N}` must contain a monochromatic arithmetic progression of length `k`. -/
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
monoAPNumber
|
/--
The **van der Waerden number**, is the smallest integer `N` such that any `r`-coloring of
`{1, ..., N}` is guaranteed to contain a monochromatic arithmetic progression of
length `k`. It is defined as the infimum of the (non-empty) set of all such numbers `N`.
-/
|
W : ℕ → ℕ := monoAPNumber 2 @[category test, AMS 11]
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
W
|
/--
An abbreviation for the van der Waerden number for 2 colors, commonly written as `W(k)`.
This represents the smallest integer `N` such that any 2-coloring of `{1, ..., N}`
must contain a monochromatic arithmetic progression of length `k`.
-/
|
monoAPNumber_two_one : W 1 = 1 := by sorry @[category test, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
monoAPNumber_two_one
|
/--
An abbreviation for the van der Waerden number for 2 colors, commonly written as `W(k)`.
This represents the smallest integer `N` such that any 2-coloring of `{1, ..., N}`
must contain a monochromatic arithmetic progression of length `k`.
-/
|
monoAPNumber_two_two : W 2 = 3 := by sorry /-- In [Er80] Erdős asks whether $$ \lim_{k \to \infty} (W(k))^{1/k} = \infty $$ -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
monoAPNumber_two_two
|
/--
An abbreviation for the van der Waerden number for 2 colors, commonly written as `W(k)`.
This represents the smallest integer `N` such that any 2-coloring of `{1, ..., N}`
must contain a monochromatic arithmetic progression of length `k`.
-/
|
erdos_138 : answer(sorry) ↔ atTop.Tendsto (fun k => (W k : ℝ)^(1/(k : ℝ))) atTop := by sorry /-- When $p$ is prime Berlekamp [Be68] has proved $W(p+1) ≥ p^{2^p}$. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
erdos_138
|
/--
In [Er80] Erdős asks whether
$$ \lim_{k \to \infty} (W(k))^{1/k} = \infty $$
-/
|
erdos_138.variants.prime (p : ℕ) (hp : p.Prime) : p * (2 ^ p) ≤ W (p + 1) := by sorry /-- Gowers [Go01] has proved $$W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}.$$ -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
erdos_138.variants.prime
|
/--
When $p$ is prime Berlekamp [Be68] has proved $W(p+1) ≥ p^{2^p}$.
-/
|
erdos_138.variants.upper (k : ℕ) : W k ≤ 2 ^ (2 ^ (2 ^ 2 ^ 2 ^ (k + 9))) := by sorry /-- In [Er81] Erdős asks whether $\frac{W(k+1)}{W(k)} \to \infty$. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
erdos_138.variants.upper
|
/--
Gowers [Go01] has proved $$W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}.$$
-/
|
erdos_138.variants.quotient : answer(sorry) ↔ atTop.Tendsto (fun k => ((W (k + 1) : ℚ)/(W k))) atTop := by sorry /-- In [Er81] Erdős asks whether $W(k+1) - W(k) \to \infty$. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
erdos_138.variants.quotient
|
/--
In [Er81] Erdős asks whether $\frac{W(k+1)}{W(k)} \to \infty$.
-/
|
erdos_138.variants.difference : answer(sorry) ↔ atTop.Tendsto (fun k => (W (k + 1) - W k)) atTop := by sorry /-- In [Er80] Erdős asks whether $W(k)/2^k\to \infty$. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
erdos_138.variants.difference
|
/--
In [Er81] Erdős asks whether $W(k+1) - W(k) \to \infty$.
-/
|
erdos_138.variants.dvd_two_pow : answer(sorry) ↔ atTop.Tendsto (fun k => ((W k : ℚ)/ (2 ^ k))) atTop := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/138.lean
|
erdos_138.variants.dvd_two_pow
|
/--
In [Er80] Erdős asks whether $W(k)/2^k\to \infty$.
-/
|
r := Set.IsAPOfLengthFree.maxCard /-- **Erdős Problem 139**: Let $r_k(N)$ be the size of the largest subset of ${1,...,N}$ which does not contain a non-trivial $k$-term arithmetic progression. Prove that $r_k(N) = o(N)$. -/ @[category research solved, AMS 5 11]
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/139.lean
|
r
| null |
erdos_139 (k : ℕ) (hk : 1 < k) : Filter.Tendsto (fun N => (r k N / N : ℝ)) Filter.atTop (𝓝 0) := by sorry /- TODO(lezeau): add the various known bounds as variants. -/
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/139.lean
|
erdos_139
|
/--
**Erdős Problem 139**:
Let $r_k(N)$ be the size of the largest subset of ${1,...,N}$ which does not contain a non-trivial
$k$-term arithmetic progression. Prove that $r_k(N) = o(N)$.
-/
|
allUniqueSums (A : Set ℕ) : Set ℕ := { n | ∃! (a : ℕ × ℕ), a.1 ≤ a.2 ∧ a.1 ∈ A ∧ a.2 ∈ A ∧ a.1 + a.2 = n } /-- The number of integers in $\{1,\ldots,N\}$ which are not representable in exactly one way as the sum of two elements from $A$ (either because they are not representable at all, or because they are representable in more than one way). -/
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/14.lean
|
allUniqueSums
| null |
nonUniqueSumCount (A : Set ℕ) (N : ℕ) : ℝ := ((Set.Icc 1 N) \ (allUniqueSums A)).ncard
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/14.lean
|
nonUniqueSumCount
|
/--
The number of integers in $\{1,\ldots,N\}$ which are not representable in exactly one way
as the sum of two elements from $A$ (either because they are not representable at all, or
because they are representable in more than one way).
-/
|
almostSquareRoot (ε : ℝ) (N : ℕ) : ℝ := N ^ (1/2 - ε)
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/14.lean
|
almostSquareRoot
|
/--
The number of integers in $\{1,\ldots,N\}$ which are not representable in exactly one way
as the sum of two elements from $A$ (either because they are not representable at all, or
because they are representable in more than one way).
-/
|
squareRoot (N : ℕ) : ℝ := Real.sqrt N /-- Let $A ⊆ \mathbb{N}$. Let $B ⊆ \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$. Is it true that for all $\epsilon > 0$ and large $N$, $|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}$? -/ @[category research open, AMS 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/14.lean
|
squareRoot
|
/--
The number of integers in $\{1,\ldots,N\}$ which are not representable in exactly one way
as the sum of two elements from $A$ (either because they are not representable at all, or
because they are representable in more than one way).
-/
|
erdos_14a : answer(sorry) ↔ ∀ A, ∀ ε > 0, nonUniqueSumCount A ≫ almostSquareRoot ε := by sorry /-- Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$? -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/14.lean
|
erdos_14a
|
/--
Let $A ⊆ \mathbb{N}$. Let $B ⊆ \mathbb{N}$ be the set of integers which are representable
in exactly one way as the sum of two elements from $A$. Is it true that for all
$\epsilon > 0$ and large $N$, $|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}$?
-/
|
erdos_14b : answer(sorry) ↔ ∃ (A : Set ℕ), IsLittleO atTop (nonUniqueSumCount A) squareRoot := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/14.lean
|
erdos_14b
|
/--
Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$?
-/
|
Set.IsPrimeProgressionOfLength (s : Set ℕ) (l : ℕ∞) : Prop := ∃ a, ENat.card s = l ∧ s = {(a + n).nth Nat.Prime | (n : ℕ) (_ : n < l)} open Nat Erdos141 /-- The first three odd primes are an example of three consecutive primes. -/ @[category test, AMS 5 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
Set.IsPrimeProgressionOfLength
|
/--
The predicate that a set `s` consists of `l` consecutive primes (possibly infinite).
This predicate does not assert a specific value for the first term.
-/
|
first_three_odd_primes : ({3, 5, 7} : Set ℕ).IsPrimeProgressionOfLength 3 := by use 1 constructor · aesop · norm_num [exists_lt_succ, or_assoc, eq_comm, Set.insert_def, show (2).nth Nat.Prime = 5 from nth_count prime_five, show (3).nth Nat.Prime = 7 from Nat.nth_count (by decide : (7).Prime)] /-- The predicate that a set `s` is both an arithmetic progression of length `l` and a progression of `l` consecutive primes. -/
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
first_three_odd_primes
|
/--
The first three odd primes are an example of three consecutive primes.
-/
|
Set.IsAPAndPrimeProgressionOfLength (s : Set ℕ) (l : ℕ) := s.IsAPOfLength l ∧ s.IsPrimeProgressionOfLength l /-- There are 3 consecutive primes in arithmetic progression. -/ @[category test, AMS 5 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
Set.IsAPAndPrimeProgressionOfLength
|
/--
The predicate that a set `s` is both an arithmetic progression of length `l` and a progression
of `l` consecutive primes.
-/
|
exists_three_consecutive_primes_in_ap : ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength 3 := by use {3, 5, 7} constructor · use 3, 2 unfold Set.IsAPOfLengthWith constructor · aesop · norm_num [exists_lt_succ, or_assoc, eq_comm, Set.insert_def] · exact first_three_odd_primes /-- Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression? -/ @[category research open, AMS 5 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
exists_three_consecutive_primes_in_ap
|
/--
There are 3 consecutive primes in arithmetic progression.
-/
|
erdos_141 : answer(sorry) ↔ ∀ k ≥ 3, ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength k := by sorry /-- The existence of such progressions has been verified for $k≤10$. -/ @[category research solved, AMS 5 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
erdos_141
|
/--
Let $k≥3$. Are there $k$ consecutive primes in arithmetic progression?
-/
|
erdos_141.variant.first_cases : (∀ k ≥ 3, k ≤ 10 → ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength k) := by sorry /-- Are there $11$ consecutive primes in arithmetic progression? -/ @[category research open, AMS 5 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
erdos_141.variant.first_cases
|
/--
The existence of such progressions has been verified for $k≤10$.
-/
|
erdos_141.variant.eleven : answer(sorry) ↔ ∃ (s : Set ℕ), s.IsAPAndPrimeProgressionOfLength 11 := by sorry /-- The set of arithmetic progressions of consecutive primes of length $k$. -/
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
erdos_141.variant.eleven
|
/--
Are there $11$ consecutive primes in arithmetic progression?
-/
|
consecutivePrimeArithmeticProgressions (k : ℕ) : Set (Set ℕ) := {s | s.IsAPAndPrimeProgressionOfLength k} /-- It is open, even for $k=3$, whether there are infinitely many such progressions. -/ @[category research open, AMS 5 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
consecutivePrimeArithmeticProgressions
|
/--
The set of arithmetic progressions of consecutive primes of length $k$.
-/
|
erdos_141.variant.infinite_three : answer(sorry) ↔ (consecutivePrimeArithmeticProgressions 3).Infinite := sorry /-- Fix a $k \geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$? -/ @[category research open, AMS 5 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
erdos_141.variant.infinite_three
|
/--
It is open, even for $k=3$, whether there are infinitely many such progressions.
-/
|
erdos_141.variant.infinite_general_case : answer(sorry) ↔ ∀ k ≥ 3, (consecutivePrimeArithmeticProgressions k).Infinite := sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/141.lean
|
erdos_141.variant.infinite_general_case
|
/--
Fix a $k \geq 3$. Is it true that there are infinitely many arithmetic prime progressions of length $k$?
-/
|
r := Set.IsAPOfLengthFree.maxCard /-- Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression. -/ @[category research open, AMS 11]
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/142.lean
|
r
| null |
erdos_142 (k : ℕ) : (fun N => (r k N : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by sorry /-- Show that $r_k(N) = o_k(N / \log N)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/142.lean
|
erdos_142
|
/--
Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset
of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.
-/
|
erdos_142.variants.lower (k : ℕ) (hk : 1 < k) : (fun N => (r k N : ℝ)) =o[atTop] (fun N : ℕ => N / (N : ℝ).log) := by sorry /-- Find functions $f_k$, such that $r_k(N) = O_k(f_k)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/142.lean
|
erdos_142.variants.lower
|
/--
Show that $r_k(N) = o_k(N / \log N)$, where $r_k(N)$ the largest possible size of a subset
of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.
-/
|
erdos_142.variants.upper (k : ℕ) : (fun N => (r k N : ℝ)) =O[atTop] (answer(sorry) : ℕ → ℝ) := by sorry -- TODO(firsching): at known upper bounds for small k /-- Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression. -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/142.lean
|
erdos_142.variants.upper
|
/--
Find functions $f_k$, such that $r_k(N) = O_k(f_k)$, where $r_k(N)$ the largest possible size of a
subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.
-/
|
erdos_142.variants.three : (fun N => (r 3 N : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/142.lean
|
erdos_142.variants.three
|
/--
Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset
of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression.
-/
|
WellSeparatedSet (A : Set ℝ) : Prop := (A ⊆ (Set.Ioi (1 : ℝ))) ∧ Set.Infinite A ∧ Set.Countable A ∧ (∀ x ∈ A, ∀ y ∈ A, x ≠ y → (∀ k ≥ (1 : ℕ), 1 ≤ |k * x - y|)) /-- Does this imply that $$ \liminf \frac{|A \cap [1,x]|}{x} = 0? $$ -/ @[category research open, AMS 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/143.lean
|
WellSeparatedSet
|
/--
Let $A \subseteq (1, \infty)$ be a countably infinite set such that for all $x\neq y\in A$ and
integers $k \geq 1$ we have $|kx - y| \geq 1$.
-/
|
erdos_143.parts.i : answer(sorry) ↔ ∀ (A : Set ℝ), WellSeparatedSet A → liminf (fun x => (A ∩ (Set.Icc 1 x)).ncard / x) atTop = 0 := by sorry /-- Or $$ \sum_{x \in A} \frac{1}{x \log x} < \infty, $$ -/ @[category research open, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/143.lean
|
erdos_143.parts.i
|
/--
Does this imply that
$$
\liminf \frac{|A \cap [1,x]|}{x} = 0?
$$
-/
|
erdos_143.parts.ii (A : Set ℝ) (h : WellSeparatedSet A) : Summable fun (x : A) ↦ 1 / (x * Real.log x) := by sorry -- TODO(firsching): add the two other conjectures. /- $$ \sum_{\substack{x < n \\ x \in A}} \frac{1}{x} = o(\log n)? $$ Perhaps even $$ \sum_{\substack{x < n \\ x \in A}} \frac{1}{x} \ll \frac{\log x}{\sqrt{\log \log x}}? $$ -/
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/143.lean
|
erdos_143.parts.ii
|
/--
Or
$$
\sum_{x \in A} \frac{1}{x \log x} < \infty,
$$
-/
|
s (n : ℕ) : ℕ := Nat.nth Squarefree n /-- Let $A(x)$ denote the set of indices $n$ for which $s_n \leq x$. -/
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/145.lean
|
s
|
/-- Let $s_1 < s_2 < \cdots$ be the sequence of squarefree numbers. -/
|
A (x : ℝ) : Finset ℕ := (Finset.Icc 0 ⌊x⌋₊).preimage s (Nat.nth_injective Nat.squarefree_infinite).injOn /-- Let $s_1 < s_2 < \cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha\geq 0$, $$ \lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha $$ exists? -/ @[category research open, AMS 11]
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/145.lean
|
A
|
/-- Let $A(x)$ denote the set of indices $n$ for which $s_n \leq x$. -/
|
erdos_145 : answer(sorry) ↔ ∀ α ≥ (0 : ℝ), ∃ β : ℝ, atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β) := by sorry /-- Erdős [Er51] proved this for all $0\leq \alpha\leq 2$. [Er51] Erdös, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/145.lean
|
erdos_145
|
/--
Let $s_1 < s_2 < \cdots$ be the sequence of squarefree numbers. Is it true that, for any
$\alpha\geq 0$,
$$
\lim_{x\to\infty} \frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha
$$
exists?
-/
|
erdos_145.variants.le_two {α : ℝ} (hα : α ∈ Set.Icc 0 2) : ∃ β : ℝ, atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β) := by sorry /-- Hooley [Ho73] extended this to all $0 \leq \alpha\leq 3$. [Ho73] Hooley, Christopher, On the intervals between consecutive terms of sequences. Proc. Symp. Pure Math, vol. 24, pp. 129-140. 1973. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/145.lean
|
erdos_145.variants.le_two
|
/--
Erdős [Er51] proved this for all $0\leq \alpha\leq 2$.
[Er51] Erdös, P., Some problems and results in elementary number theory.
Publ. Math. Debrecen (1951), 103-109.
-/
|
erdos_145.variants.le_three {α : ℝ} (hα : α ∈ Set.Icc 0 3) : ∃ β : ℝ, atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β) := by sorry /-- Greaves, Harman, and Huxley [GHH97] showed that this is true for $0 \leq \alpha\leq 11/3$. [GHH97] Greaves, G. R. H. and Harman, G. and Huxley, M. N., Sieve Methods, Exponential Sums, and their Applications in Number Theory. (1997). -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/145.lean
|
erdos_145.variants.le_three
|
/--
Hooley [Ho73] extended this to all $0 \leq \alpha\leq 3$.
[Ho73] Hooley, Christopher, On the intervals between consecutive terms of sequences. Proc. Symp. Pure Math, vol. 24, pp. 129-140. 1973.
-/
|
erdos_145.variants.le_eleven_thirds {α : ℝ} (hα : α ∈ Set.Icc 0 (11 / 3)) : ∃ β : ℝ, atTop.Tendsto (fun x : ℝ ↦ 1 / x * ∑ n ∈ A x, (s (n + 1) - s n : ℝ) ^ α) (𝓝 β) := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/145.lean
|
erdos_145.variants.le_eleven_thirds
|
/--
Greaves, Harman, and Huxley [GHH97] showed that this is true for $0 \leq \alpha\leq 11/3$.
[GHH97] Greaves, G. R. H. and Harman, G. and Huxley, M. N., Sieve Methods, Exponential Sums, and
their Applications in Number Theory. (1997).
-/
|
f (n : ℕ) : ℕ := ⨅ A : {A : Set ℕ | A.ncard = n ∧ IsSidon A}, {s : ℕ | s - 1 ∉ A.1 + A.1 ∧ s ∈ A.1 + A.1 ∧ s + 1 ∉ A.1 + A.1}.ncard /-- Must `lim f n = ∞`? -/ @[category research open, AMS 5]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/152.lean
|
f
|
/-- Define `f n` to be the minimum of `|{s | s - 1 ∉ A + A, s ∈ A + A, s + 1 ∉ A + A}|` as `A`
ranges over all Sidon sets of size `n`. -/
|
erdos_152 : answer(sorry) ↔ Tendsto f atTop atTop := by sorry /-- Must `f n ≫ n ^ 2`? -/ @[category research open, AMS 5]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/152.lean
|
erdos_152
|
/-- Must `lim f n = ∞`? -/
|
erdos_152.square : answer(sorry) ↔ (fun n => f n : ℕ → ℝ) ≫ (fun n => n ^ 2 : ℕ → ℝ) := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/152.lean
|
erdos_152.square
|
/-- Must `f n ≫ n ^ 2`? -/
|
f (n : ℕ) : ℝ := ⨅ A : {A : Finset ℕ | A.card = n ∧ IsSidon A}, let s := (A.1 + A).orderIsoOfFin rfl ∑ i : Set.Ico 1 ((A.1 + A).card), (s ⟨i, i.2.2⟩ - s ⟨i - 1, by grind⟩) ^ 2 / (n : ℝ) /-- Must `lim f n = ∞`? -/ @[category research open, AMS 5]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/153.lean
|
f
|
/-- Define `f n` to be the minimum of
`∑ (i : Set.Ico 1 ((A + A).card), (s i - s (i - 1)) ^ 2 / n` as `A` ranges over all Sidon sets
of size `n`, where `s` is an order embedding from `Fin n` into `A`. -/
|
erdos_153 : answer(sorry) ↔ Tendsto f atTop atTop := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/153.lean
|
erdos_153
|
/-- Must `lim f n = ∞`? -/
|
F (N : ℕ) : ℕ := Finset.maxSidonSubsetCard (Finset.Icc 1 N) /-- Is it true that for every $k \geq 1$ we have $$ F(N + k) \leq F(N) + 1 $$ for all sufficiently large $N$? -/ @[category research open, AMS 5]
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/155.lean
|
F
|
/--
Let $F(N)$ be the size of the largest Sidon subset of $\{1, \dots, N\}$.
-/
|
erdos_155 : answer(sorry) ↔ ∀ k ≥ 1, ∀ᶠ N in atTop, F (N + k) ≤ F N + 1 := by sorry -- TODO: This may even hold with $k \approx ε * N ^ (1 / 2)$.
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/155.lean
|
erdos_155
|
/--
Is it true that for every $k \geq 1$ we have
$$
F(N + k) \leq F(N) + 1
$$
for all sufficiently large $N$?
-/
|
B2 (g : ℕ) (A : Set ℕ) : Prop := ∀ n, {x : ℕ × ℕ | x.1 + x.2 = n ∧ x.1 ≤ x.2 ∧ x.1 ∈ A ∧ x.2 ∈ A}.encard ≤ g /-- A set is `B₂[1]` iff it is Sidon. -/ @[category API, AMS 5, simp]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/158.lean
|
B2
|
/-- A set `A ⊆ ℕ` is said to be a `B₂[g]` set if for all `n`, the equation
`a + a' = n, a ≤ a', a, a' ∈ A` has at most `g` solutions. This is defined in [ESS94]. -/
|
b2_one {A : Set ℕ} : B2 1 A ↔ IsSidon A where mp hA a₁ ha₁ a₂ ha₂ b₁ hb₁ b₂ hb₂ h := by wlog h₁ : a₁ ≤ b₁ · have := this hA _ hb₁ _ ha₂ _ ha₁ _ hb₂ grind wlog h₂ : a₂ ≤ b₂ · have := this hA _ ha₁ _ hb₂ _ hb₁ _ ha₂ grind have := Set.encard_le_one_iff.1 (hA (a₁ + b₁)) ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (by simp [*]) (by simp [*]) grind mpr hA n := by refine Set.encard_le_one_iff.2 fun x y ⟨h, p, q⟩ ⟨r, s, t⟩ => ?_ have := hA x.1 q.1 y.1 t.1 x.2 q.2 y.2 t.2 (h.trans r.symm) grind namespace Erdos158 /-- Let `A` be an infinite `B₂[2]` set. Must `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0`? -/ @[category research open, AMS 5]
|
lemma
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/158.lean
|
b2_one
|
/-- A set is `B₂[1]` iff it is Sidon. -/
|
erdos_158.B22 : answer(sorry) ↔ ∀ A : Set ℕ, A.Infinite → B2 2 A → liminf (fun N : ℕ => (A ∩ .Iio N).ncard * (N : ℝ) ^ (- 1 / 2 : ℝ)) atTop = 0 := by sorry /-- Let `A` be an infinite Sidon set. Then `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) * (log N) ^ (1 / 2) < ∞`. This is proved in [ESS94]. -/ @[category research solved, AMS 5]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/158.lean
|
erdos_158.B22
|
/-- Let `A` be an infinite `B₂[2]` set. Must `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0`? -/
|
erdos_158.isSidon' {A : Set ℕ} (hAinf : A.Infinite) (hAsid : IsSidon A) : liminf (fun N ↦ ENNReal.ofReal ((A ∩ .Iio N).ncard * N ^ (- 1 / 2 : ℝ) * log N ^ (1 / 2 : ℝ))) atTop < ⊤ := by sorry /-- As a corollary of `erdos_158.isSidon'`, we can prove that `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0` for any infinite Sidon set `A`. -/ @[category research solved, AMS 5]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/158.lean
|
erdos_158.isSidon'
|
/-- Let `A` be an infinite Sidon set. Then
`liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) * (log N) ^ (1 / 2) < ∞`. This is proved in [ESS94]. -/
|
erdos_158.isSidon {A : Set ℕ} (hAinf : A.Infinite) (hAsid : IsSidon A) : liminf (fun N : ℕ => (A ∩ .Iio N).ncard * (N : ℝ) ^ (- 1 / 2 : ℝ)) atTop = 0 := by have := erdos_158.isSidon' hAinf hAsid contrapose! this with h rw [Tendsto.liminf_eq] refine ENNReal.tendsto_ofReal_atTop.comp ?_ obtain ⟨c, hc_pos, hc⟩ : ∃ c > (0 : ℝ), ∀ᶠ N in atTop, c ≤ (A ∩ .Iio N).ncard * N ^ (- 1 / 2 : ℝ) := by suffices ∃ a ∈ {a | ∃ c : ℕ, ∀ b ≥ c, a ≤ ↑(A ∩ .Iio b).ncard * (b : ℝ) ^ (-1 / 2 : ℝ)}, 0 < a by aesop by_contra! ha simp only [liminf_eq, eventually_atTop] at h exact h <| le_antisymm (csSup_le ⟨0, 0, fun n hn => by positivity⟩ ha) <| (le_csSup ⟨0, ha⟩ ⟨0, fun n hn => by positivity⟩) refine tendsto_atTop_mono' atTop (f₁ := fun N : ℕ => c * log N ^ (1 / 2 : ℝ)) ?_ ?_ · filter_upwards [hc] with n hn grw [hn] · refine .const_mul_atTop hc_pos ?_ simpa using (tendsto_rpow_atTop (by linarith : 0 < 1 / (2 : ℝ))).comp (Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop)
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/158.lean
|
erdos_158.isSidon
|
/-- As a corollary of `erdos_158.isSidon'`, we can prove that
`liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0` for any infinite Sidon set `A`. -/
|
erdos_160.h (n : ℕ) : ℕ := sInf {k | ∃ (colouring : Finset.Icc 1 n → Fin k), ∀ (progression : Set ℕ), (progression ⊆ Finset.Icc 1 n ∧ progression.IsAPOfLength 4) → 3 ≤ (colouring '' {k | (k : ℕ) ∈ progression}).ncard} open Filter /-- On [Mathoverflow](https://mathoverflow.net/a/410815) user [leechlattice](https://mathoverflow.net/users/125498/leechlattice) shows that $h(n) \ll n^{\frac 2 3}$. -/ @[category research solved, AMS 5 51]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/160.lean
|
erdos_160.h
|
/--
Let $h(n)$ be the smallest $k$ such that $\{1,\ldots,n\}$ can be coloured with $k$ colours
so that every four-term arithmetic progression must contain at least three distinct colours.
-/
|
erdos_160.known_upper : (fun n => (erdos_160.h n : ℝ)) =O[atTop] fun n => (n : ℝ) ^ ((2 : ℝ) / 3) := by sorry open Real /-- The observation of Zachary Hunter in [that question](https://mathoverflow.net/q/410808) coupled with the bounds of Kelley-Meka [KeMe23](https://arxiv.org/abs/2302.05537) imply that $$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$ for some $c > 0$. -/ @[category research solved, AMS 5 51]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/160.lean
|
erdos_160.known_upper
|
/--
On [Mathoverflow](https://mathoverflow.net/a/410815) user
[leechlattice](https://mathoverflow.net/users/125498/leechlattice) shows that
$h(n) \ll n^{\frac 2 3}$.
-/
|
erdos_160.known_lower : ∃ c > 0, (fun (n : ℕ) => exp (c * log (n : ℝ) ^ ((1 : ℝ) / 12))) =O[atTop] fun n => (erdos_160.h n : ℝ):= by sorry /-- Estimate $h(n)$ by finding a better upper bound. -/ @[category research open, AMS 5 51]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/160.lean
|
erdos_160.known_lower
|
/--
The observation of Zachary Hunter in [that question](https://mathoverflow.net/q/410808)
coupled with the bounds of Kelley-Meka [KeMe23](https://arxiv.org/abs/2302.05537) imply that
$$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$
for some $c > 0$.
-/
|
erdos_160.better_upper : let upper_bound : ℕ → ℝ := answer(sorry) (fun n => (erdos_160.h n : ℝ)) =O[atTop] upper_bound ∧ upper_bound =o[atTop] fun n => (n : ℝ) ^ ((2 : ℝ) / 3) := by sorry /-- Estimate $h(n)$ by finding a better lower bound. -/ @[category research open, AMS 5 51]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/160.lean
|
erdos_160.better_upper
|
/--
Estimate $h(n)$ by finding a better upper bound.
-/
|
erdos_160.better_lower : let lower_bound : ℕ → ℝ := answer(sorry) (lower_bound =O[atTop] fun n => (erdos_160.h n : ℝ)) ∧ ∀ c > 0, (fun (n : ℕ) => exp (c * log n ^ ((1 : ℝ) / 12))) =O[atTop] (fun n => (erdos_160.h n : ℝ)) → ∀ c > 0, (fun (n : ℕ) => exp (c * log n ^ ((1 : ℝ) / 12))) =o[atTop] lower_bound := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/160.lean
|
erdos_160.better_lower
|
/--
Estimate $h(n)$ by finding a better lower bound.
-/
|
NonTernary (S : Finset ℕ) : Prop := ∀ n : ℕ, n ∉ S ∨ 2*n ∉ S ∨ 3*n ∉ S /--`IntervalNonTernarySets N` is the (fin)set of non ternary subsets of `{1,...,N}`. The advantage of defining it as below is that some proofs (e.g. that of `F 3 = 2`) become `rfl`.-/
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
NonTernary
|
/--Say a finite set of natural numbers is *non ternary* if it contains no
3-term arithmetic progression of the form `n, 2n, 3n`.-/
|
IntervalNonTernarySets (N : ℕ) : Finset (Finset ℕ) := (Finset.Icc 1 N).powerset.filter fun S => ∀ n ∈ Finset.Icc 1 (N / 3 : ℕ), n ∉ S ∨ 2*n ∉ S ∨ 3*n ∉ S /--`F N` is the size of the largest non ternary subset of `{1,...,N}`.-/
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
IntervalNonTernarySets
|
/--`IntervalNonTernarySets N` is the (fin)set of non ternary subsets of `{1,...,N}`.
The advantage of defining it as below is that some proofs (e.g. that of `F 3 = 2`) become `rfl`.-/
|
F (N : ℕ) : ℕ := (IntervalNonTernarySets N).sup Finset.card @[category API, AMS 5 11]
|
abbrev
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
F
|
/--`F N` is the size of the largest non ternary subset of `{1,...,N}`.-/
|
F_0 : F 0 = 0 := rfl @[category API, AMS 5 11]
|
lemma
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
F_0
|
/--`F N` is the size of the largest non ternary subset of `{1,...,N}`.-/
|
F_1 : F 1 = 1 := rfl @[category API, AMS 5 11]
|
lemma
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
F_1
|
/--`F N` is the size of the largest non ternary subset of `{1,...,N}`.-/
|
F_2 : F 2 = 2 := rfl @[category API, AMS 5 11]
|
lemma
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
F_2
|
/--`F N` is the size of the largest non ternary subset of `{1,...,N}`.-/
|
F_3 : F 3 = 2 := rfl /-- Sanity check: elements of `IntervalNonTernarySets N` are precisely non ternary subsets of `{1,...,N}` -/ @[category API, AMS 5 11]
|
lemma
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
F_3
|
/--`F N` is the size of the largest non ternary subset of `{1,...,N}`.-/
|
mem_IntervalNonTernarySets_iff (N : ℕ) (S : Finset ℕ) : S ∈ IntervalNonTernarySets N ↔ NonTernary S ∧ S ⊆ Finset.Icc 1 N := by refine ⟨fun h => ?_, fun h => by simpa [h, IntervalNonTernarySets] using fun _ _ _ => h.1 _⟩ simp_all [NonTernary, IntervalNonTernarySets, S.subset_iff, Nat.le_div_iff_mul_le, mul_comm, or_iff_not_imp_left] exact fun n hn₁ hn₂ hn₃ => h.2 n (h.1 hn₁).1 (h.1 hn₃).2 hn₁ hn₂ hn₃ /-- Sanity check: if `S` is a maximal non ternary subset of `{1,..., N}` then `F N` is given by the cardinality of `S` -/ @[category API, AMS 5 11]
|
lemma
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
mem_IntervalNonTernarySets_iff
|
/--
Sanity check: elements of `IntervalNonTernarySets N` are precisely non ternary subsets of
`{1,...,N}`
-/
|
F_eq_card (N : ℕ) (S : Finset ℕ) (hS : S ⊆ Finset.Icc 1 N) (hS' : NonTernary S) (hS'' : ∀ T, T ⊆ Finset.Icc 1 N → NonTernary T → S.card ≤ T.card → T.card = S.card) : F N = S.card := by sorry /-- What is the limit $F(N)/N$ as $N \to \infty$? -/ @[category research open, AMS 11]
|
lemma
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
F_eq_card
|
/--
Sanity check: if `S` is a maximal non ternary subset of `{1,..., N}` then `F N` is given by the
cardinality of `S`
-/
|
erdos_168.parts.i : Filter.Tendsto (fun N => (F N / N : ℝ)) Filter.atTop (𝓝 answer(sorry)) := by sorry /-- Is the limit $F(N)/N$ as $N \to \infty$ irrational? -/ @[category research open, AMS 5 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
erdos_168.parts.i
|
/-- What is the limit $F(N)/N$ as $N \to \infty$? -/
|
erdos_168.parts.ii : answer(sorry) ↔ Irrational (Filter.atTop.limsup (fun N => (F N / N : ℝ))) := by sorry /-- The limit $F(N)/N$ as $N \to \infty$ exists. (proved by Graham, Spencer, and Witsenhausen) -/ @[category research solved, AMS 5 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
erdos_168.parts.ii
|
/-- Is the limit $F(N)/N$ as $N \to \infty$ irrational? -/
|
erdos_168.variants.limit_exists : ∃ x, Filter.Tendsto (fun N => (F N / N : ℝ)) Filter.atTop (𝓝 x) := by sorry
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/168.lean
|
erdos_168.variants.limit_exists
|
/-- The limit $F(N)/N$ as $N \to \infty$ exists. (proved by Graham, Spencer, and Witsenhausen) -/
|
IsClusterPrime (p : ℕ) : Prop := p.Prime ∧ ∀ {n : ℕ}, Even n → n ≤ (p - 3 : ℤ) → ∃ q₁ q₂ : ℕ, q₁.Prime ∧ q₂.Prime ∧ q₁ ≤ p ∧ q₂ ≤ p ∧ n = (q₁ - q₂ : ℤ) /-- **Erdős Problem 17.** Are there infinitely many cluster primes? -/ @[category research open, AMS 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/17.lean
|
IsClusterPrime
|
/-- A prime $p$ is a cluster prime if every even natural number
$n \le p - 3$ can be written as a difference of two primes
$q_1 - q_2$ with $q_1, q_2 \le p$. -/
|
erdos_17 : answer(sorry) ↔ {p : ℕ | IsClusterPrime p}.Infinite := by sorry /-- The counting function of cluster primes $\le n$. -/
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/17.lean
|
erdos_17
|
/-- **Erdős Problem 17.** Are there infinitely many cluster primes? -/
|
clusterPrimeCount (n : ℕ) : ℕ := Nat.card {p : ℕ | p ≤ n ∧ IsClusterPrime p} /-- In 1999 Blecksmith, Erdős, and Selfridge [BES99] proved the upper bound $$\pi^{\mathcal{C}}(x) \ll_A x(\log x)^{-A}$$ for every real $A > 0$. [BES99] Blecksmith, Richard and Erd\H os, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43--48. -/ @[category research solved, AMS 11]
|
def
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/17.lean
|
clusterPrimeCount
|
/-- The counting function of cluster primes $\le n$. -/
|
erdos_17.variants.upper_BES {A : ℝ} (hA : 0 < A) : (fun x ↦ (clusterPrimeCount x : ℝ)) =O[atTop] fun x ↦ x / (log x) ^ A := by sorry /-- In 2003, Elsholtz [El03] refined the upper bound to $$\pi^{\mathcal{C}}(x) \ll x\,\exp\!\bigl(-c(\log\log x)^2\bigr)$$ for every real $0 < c < 1/8$. [El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284. -/ @[category research solved, AMS 11]
|
theorem
|
FormalConjectures
|
[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/17.lean
|
erdos_17.variants.upper_BES
|
/--
In 1999 Blecksmith, Erdős, and Selfridge [BES99] proved the upper bound
$$\pi^{\mathcal{C}}(x) \ll_A x(\log x)^{-A}$$ for every real $A > 0$.
[BES99] Blecksmith, Richard and Erd\H os, Paul and Selfridge, J. L., Cluster primes. Amer. Math. Monthly (1999), 43--48.
-/
|
erdos_17.variants.upper_Elsholtz : ∃ C : ℝ, 0 < C ∧ ∀ c ∈ Set.Ioo 0 (1 / 8), IsBigOWith C atTop (fun x ↦ (clusterPrimeCount x : ℝ)) (fun x ↦ x * exp (-c * (log (log x)) ^ 2)) := by sorry /-- $97$ is the smallest prime that is not a cluster prime. -/ @[category test, AMS 11]
|
theorem
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FormalConjectures
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[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/17.lean
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erdos_17.variants.upper_Elsholtz
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/--
In 2003, Elsholtz [El03] refined the upper bound to
$$\pi^{\mathcal{C}}(x) \ll x\,\exp\!\bigl(-c(\log\log x)^2\bigr)$$
for every real $0 < c < 1/8$.
[El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284.
-/
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isClusterPrime_97_isLeast_non_cluster : IsLeast {p : ℕ | p.Prime ∧ ¬ IsClusterPrime p} 97 := by sorry
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theorem
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FormalConjectures
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[
"import FormalConjectures.Util.ProblemImports"
] |
FormalConjectures/ErdosProblems/17.lean
|
isClusterPrime_97_isLeast_non_cluster
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/-- $97$ is the smallest prime that is not a cluster prime. -/
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