statement stringlengths 1 2.98k | proof stringlengths 0 7.39k | type stringclasses 10
values | symbolic_name stringlengths 1 115 | library stringclasses 88
values | filename stringclasses 870
values | imports listlengths 0 89 | deps listlengths 0 64 | docstring stringlengths 0 1.64k | source_url stringclasses 1
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erdos_1055.variants.erdos_limit :
Filter.atTop.Tendsto (fun r ↦ (p r : ℝ) ^ (1 / r : ℝ)) Filter.atTop | by
sorry | theorem | Erdos1055.erdos_1055.variants.erdos_limit | ErdosProblems | FormalConjectures/ErdosProblems/1055.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are
$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor
of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor.
If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?
Erdos conjectured t... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1055.variants.selfridge_limit :
∃ M, ∀ r, (p r : ℝ) ^ (1 / r : ℝ) ≤ M | by
sorry | theorem | Erdos1055.erdos_1055.variants.selfridge_limit | ErdosProblems | FormalConjectures/ErdosProblems/1055.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are
$2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor
of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor.
If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave?
Selfridge conjectur... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
AllModProdEqualsOne (p : ℕ) {k : ℕ} (boundaries : Fin (k + 1) → ℕ) : Prop | ∀ i : Fin k,
(∏ n ∈ Finset.Ico (boundaries i.castSucc) (boundaries (i.castSucc + 1)), n) ≡ 1 [MOD p] | def | Erdos1056.AllModProdEqualsOne | ErdosProblems | FormalConjectures/ErdosProblems/1056.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The proposition that the modular product of a collection of consecutive interval equals $1$ modulo $p$,
where intervals are defined by a function specifying the consecutive boundaries. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1056 : answer(sorry) ↔
∀ k ≥ 2, ∃ (p : ℕ) (_ : p.Prime) (boundaries : Fin (k + 1) → ℕ) (_ : StrictMono boundaries),
AllModProdEqualsOne p boundaries | by
sorry | theorem | Erdos1056.erdos_1056 | ErdosProblems | FormalConjectures/ErdosProblems/1056.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$
such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1056.variants.k2 :
AllModProdEqualsOne 11 ![3, 5, 8] | by
unfold AllModProdEqualsOne
decide | theorem | Erdos1056.erdos_1056.variants.k2 | ErdosProblems | FormalConjectures/ErdosProblems/1056.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979
Erdős observed that $3 * 4 \equiv 5 * 6 * 7 \equiv 1 \mod 11$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1056.variants.k3 :
AllModProdEqualsOne 17 ![2, 6, 12, 16] | by
unfold AllModProdEqualsOne
decide | theorem | Erdos1056.erdos_1056.variants.k3 | ErdosProblems | FormalConjectures/ErdosProblems/1056.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Makowski [Ma83] found, for $k=3$:
$2 * 3 * 4 * 5 \equiv 6 * 7 * 8 * 9 * 10 * 11 \equiv 12 * 13 * 14 * 15 \equiv 1 \mod 17$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1056.variants.noll_simmons :
answer(sorry) ↔ ∀ᶠ k in Filter.atTop,
∃ (p : ℕ) (_ : p.Prime) (Q : Fin k → ℕ) (_ : StrictMono Q) (_ : ∀ i, Q i < p),
∀ i j : Fin k, (Q i)! ≡ (Q j)! [MOD p] | by
sorry | theorem | Erdos1056.erdos_1056.variants.noll_simmons | ErdosProblems | FormalConjectures/ErdosProblems/1056.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Noll and Simmons asked, more generally, whether there are solutions to
$q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$). | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsFactorial (d : ℕ) : Prop | d ∈ Set.range Nat.factorial | def | Erdos1059.IsFactorial | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
factorialsLessThanN (n : ℕ) : Set ℕ | { d | d < n ∧ IsFactorial d } | def | Erdos1059.factorialsLessThanN | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
AllFactorialSubtractionsComposite (n : ℕ) : Prop | ∀d ∈ factorialsLessThanN n, (n - d).Composite | def | Erdos1059.AllFactorialSubtractionsComposite | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
erdos_1059 :
answer(sorry) ↔ Set.Infinite {p | p.Prime ∧ AllFactorialSubtractionsComposite p} | by
sorry | theorem | Erdos1059.erdos_1059 | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
DecidableIsFactorial (d : ℕ) : Prop | ((Finset.Icc 0 d).filter (λ k => Nat.factorial k = d)).Nonempty | abbrev | Erdos1059.DecidableIsFactorial | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
decidableFactorialsLessThanN (n : ℕ) : Finset ℕ | (Finset.range n).filter DecidableIsFactorial | def | Erdos1059.decidableFactorialsLessThanN | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
DecidableAllFactorialSubtractionsComposite (n : ℕ) : Prop | ∀ d ∈ decidableFactorialsLessThanN n, (n - d).Composite | def | Erdos1059.DecidableAllFactorialSubtractionsComposite | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
isFactorial_equivalent (d : ℕ) :
IsFactorial d ↔ DecidableIsFactorial d | by
unfold IsFactorial DecidableIsFactorial
simp
constructor
· rintro ⟨k, hk⟩
use k
rw [Finset.mem_filter]
constructor
· have hk : k <= d := by
rw [← hk]
apply Nat.self_le_factorial
rw [Finset.mem_Icc]
exact ⟨Nat.zero_le k, hk⟩
· exact hk
· rintro ⟨k, hk⟩
use... | lemma | Erdos1059.isFactorial_equivalent | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
factorialsLessThanN_equivalent (n : ℕ) :
factorialsLessThanN n = ↑(decidableFactorialsLessThanN n) | by
ext d
unfold factorialsLessThanN decidableFactorialsLessThanN
simp
exact λ _ => isFactorial_equivalent d | lemma | Erdos1059.factorialsLessThanN_equivalent | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
allFactorialSubtractionsComposite_equivalent (d : ℕ) :
DecidableAllFactorialSubtractionsComposite d ↔ AllFactorialSubtractionsComposite d | by
unfold AllFactorialSubtractionsComposite DecidableAllFactorialSubtractionsComposite
rw [factorialsLessThanN_equivalent d]
simp | lemma | Erdos1059.allFactorialSubtractionsComposite_equivalent | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
allFactorialSubtractionsComposite_101 : AllFactorialSubtractionsComposite 101 | by
have h : DecidableAllFactorialSubtractionsComposite 101 := by
norm_num [DecidableAllFactorialSubtractionsComposite, decidableFactorialsLessThanN]
decide +kernel
exact (allFactorialSubtractionsComposite_equivalent 101).mp h | theorem | Erdos1059.allFactorialSubtractionsComposite_101 | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
allFactorialSubtractionsComposite_211 : AllFactorialSubtractionsComposite 211 | by
have h : DecidableAllFactorialSubtractionsComposite 211 := by
norm_num [DecidableAllFactorialSubtractionsComposite, decidableFactorialsLessThanN]
decide +kernel
exact (allFactorialSubtractionsComposite_equivalent 211).mp h | theorem | Erdos1059.allFactorialSubtractionsComposite_211 | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
notAllFactorialSubtractionsComposite_89 : ¬(AllFactorialSubtractionsComposite 89) | by
have h : ¬(DecidableAllFactorialSubtractionsComposite 89) := by
unfold DecidableAllFactorialSubtractionsComposite decidableFactorialsLessThanN
intro h
specialize h 6
have : Nat.Prime (89 - 6) := by norm_num
contradiction
simp [allFactorialSubtractionsComposite_equivalent] at h
exact h | theorem | Erdos1059.notAllFactorialSubtractionsComposite_89 | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
testFactorialsLessThanN : factorialsLessThanN 100 = {1, 2, 6, 24} | by
have h : decidableFactorialsLessThanN 100 = {1, 2, 6, 24} := by
norm_num [decidableFactorialsLessThanN]
decide +kernel
rw [factorialsLessThanN_equivalent]
simp [h] | theorem | Erdos1059.testFactorialsLessThanN | ErdosProblems | FormalConjectures/ErdosProblems/1059.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
erdos_1060.parts.i :
∃ h : ℕ → ℝ,
h =o[atTop] (fun n ↦ 1 / log (log n)) ∧ ∀ᶠ n in atTop, #{k ≤ n | k * σ 1 k = n} ≤ (n : ℝ) ^ h n | by sorry | theorem | Erdos1060.erdos_1060.parts.i | ErdosProblems | FormalConjectures/ErdosProblems/1060.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The conjecture is about the function $f(n)$ which counts the number of solutions to
$k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower
than any power of $n^(\frac{1}{\log\log n})$. The second bound is that $f(n)$ is at most a power of
$\log n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1060.parts.ii :
∃ (C : ℝ), (fun n ↦ (#{k ≤ n | k * σ 1 k = n} : ℝ)) =O[atTop]
(fun n ↦ log n ^ C) | by sorry | theorem | Erdos1060.erdos_1060.parts.ii | ErdosProblems | FormalConjectures/ErdosProblems/1060.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Part (ii) of Erdős Problem 1060: bound on the number of $k \le n$ with $k \sigma_1(k) = n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
S (x : ℝ) : ℝ | ((Finset.Icc 1 ⌊x⌋₊ ×ˢ Finset.Icc 1 ⌊x⌋₊).filter fun (a, b) ↦
a + b ≤ x ∧ σ 1 a + σ 1 b = σ 1 (a + b)).card | abbrev | Erdos1061.S | ErdosProblems | FormalConjectures/ErdosProblems/1061.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let `S x` count the number of **ordered** pairs of positive integers `(a, b)` with `a + b ≤ x`
such that `σ(a) + σ(b) = σ(a + b)`, where `σ` is the sum of divisors function.
In particular, `(a, b)` and `(b, a)` are counted separately; an unordered variant could be obtained
by additionally requiring `a ≤ b`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1061 : answer(sorry) ↔ ∃ c : ℝ, 0 < c ∧ S ~[atTop] (fun x : ℝ ↦ c * x) | by
sorry | theorem | Erdos1061.erdos_1061 | ErdosProblems | FormalConjectures/ErdosProblems/1061.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | How many (ordered) solutions are there to `σ(a) + σ(b) = σ(a + b)` with `a + b ≤ x`?
Is it true that this number is asymptotic to `c * x` for some constant `c > 0`? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
ForkFree (A : Set ℕ) : Prop | ∀ a ∈ A, ({b | b ∈ A \ {a} ∧ a ∣ b} : Set ℕ).Subsingleton | def | Erdos1062.ForkFree | ErdosProblems | FormalConjectures/ErdosProblems/1062.lean | [
"FormalConjectures.Util.ProblemImports",
"Mathlib.Topology.Basic"
] | [] | A set `A` of positive integers is fork-free if no element divides two distinct
other elements of `A`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f (n : ℕ) : ℕ | Nat.findGreatest (fun k => ∃ A ⊆ Set.Icc 1 n, ForkFree A ∧ A.ncard = k) n | def | Erdos1062.f | ErdosProblems | FormalConjectures/ErdosProblems/1062.lean | [
"FormalConjectures.Util.ProblemImports",
"Mathlib.Topology.Basic"
] | [] | The extremal function from Erdős problem 1062: the largest size of a fork-free subset of
`{1,...,n}`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1062.parts.ii :
(∃ l, Tendsto (fun n => (f n : ℝ) / n) atTop (𝓝 l) ∧ Irrational l) ↔ answer(sorry) | by
sorry | theorem | Erdos1062.erdos_1062.parts.ii | ErdosProblems | FormalConjectures/ErdosProblems/1062.lean | [
"FormalConjectures.Util.ProblemImports",
"Mathlib.Topology.Basic"
] | [] | Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is
irrational. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1062.variants.lower_bound (n : ℕ) : ⌈(2 * n / 3 : ℝ)⌉₊ ≤ f n | by
classical
set b : ℕ := n / 3 with hb
let A : Finset ℕ := .Icc (b + 1) n
calc
⌈(2 * n / 3 : ℝ)⌉₊
≤ n - b := by
grw [Nat.ceil_le, Nat.cast_sub (by omega), le_sub_iff_add_le, hb, Nat.cast_div_le]
-- FIXME: `ring` should have some basic inequality support.
apply le_of_eq
ring
... | theorem | Erdos1062.erdos_1062.variants.lower_bound | ErdosProblems | FormalConjectures/ErdosProblems/1062.lean | [
"FormalConjectures.Util.ProblemImports",
"Mathlib.Topology.Basic"
] | [] | The interval `[⌊n/3⌋, n]` is fork-free, and therefore `f n` is at least `⌈2n / 3⌉`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1062.variants.lebensold_bounds :
∀ᶠ n in atTop, (0.6725 : ℝ) * n ≤ f n ∧ f n ≤ (0.6736 : ℝ) * n | by
sorry | theorem | Erdos1062.erdos_1062.variants.lebensold_bounds | ErdosProblems | FormalConjectures/ErdosProblems/1062.lean | [
"FormalConjectures.Util.ProblemImports",
"Mathlib.Topology.Basic"
] | [] | Lebensold proved that for large `n`, the function `f n` lies between `0.6725 n` and
`0.6736 n`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
n (k : ℕ) : ℕ | sInf {m | 2 * k ≤ m ∧ ∃ i0 < k, ¬ (m - i0) ∣ m.choose k ∧
∀ i < k, i ≠ i0 → (m - i) ∣ m.choose k} | def | Erdos1063.n | ErdosProblems | FormalConjectures/ErdosProblems/1063.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $n_k$ be the least $n \ge 2k$ such that all but one of the integers $n - i$ with
$0 \le i < k$ divide $\binom{n}{k}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1063.better_upper :
let upper_bound : ℕ → ℝ | answer(sorry)
(fun k => (n k : ℝ)) =O[atTop] upper_bound ∧
upper_bound =o[atTop] fun k =>
(k : ℝ) * ((Finset.Icc 1 (k - 1)).lcm (fun n : ℕ => n) : ℝ) := by
sorry | theorem | Erdos1063.erdos_1063.better_upper | ErdosProblems | FormalConjectures/ErdosProblems/1063.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Estimate $n_k$ by finding a better upper bound. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1063.variants.exists_exception {n k : ℕ} (hk : 2 ≤ k) (h : 2 * k ≤ n) :
∃ i < k, ¬ (n - i) ∣ n.choose k | by
sorry | theorem | Erdos1063.erdos_1063.variants.exists_exception | ErdosProblems | FormalConjectures/ErdosProblems/1063.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős and Selfridge noted that, for $n \ge 2k$ with $k \ge 2$, at least one of the numbers
$n - i$ for $0 \le i < k$ fails to divide $\binom{n}{k}$ ([ErSe83]). | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1063.variants.small_values :
n 2 = 4 ∧ n 3 = 6 ∧ n 4 = 9 ∧ n 5 = 12 | by
refine ⟨?_, ?_, ?_, ?_⟩
· -- n 2 = 4 : every element of the set is ≥ 2 * 2 = 4, and 4 itself lies in the set
apply le_antisymm
· exact Nat.sInf_le (by decide)
· apply le_csInf ⟨4, by decide⟩
rintro b hb
have : 2 * 2 ≤ b := hb.1
omega
· -- n 3 = 6
apply le_antisymm
· exact ... | theorem | Erdos1063.erdos_1063.variants.small_values | ErdosProblems | FormalConjectures/ErdosProblems/1063.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The initial values satisfy $n_2 = 4$, $n_3 = 6$, $n_4 = 9$, and $n_5 = 12$ ([Gu04], Problem B31). | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1063.variants.monier_upper_bound {k : ℕ} (hk : 3 ≤ k) :
n k ≤ k ! | by
sorry | theorem | Erdos1063.erdos_1063.variants.monier_upper_bound | ErdosProblems | FormalConjectures/ErdosProblems/1063.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Monier observed that $n_k \le k!$ for $k \ge 3$ ([Mo85]).
TODO: Find reference | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1063.variants.cambie_upper_bound {k : ℕ} (hk : 3 ≤ k) :
n k ≤ k * (Finset.Icc 1 (k - 1)).lcm id | by
sorry | theorem | Erdos1063.erdos_1063.variants.cambie_upper_bound | ErdosProblems | FormalConjectures/ErdosProblems/1063.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | [Cambie observed](https://www.erdosproblems.com/1063) the improved bound
$n_k \le k \cdot \operatorname{lcm}(1, \dotsc, k - 1)$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1063.variants.exp_upper_bound :
∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧
∀ k, (n k : ℝ) ≤ exp ((1 + f k) * k) | by
sorry | theorem | Erdos1063.erdos_1063.variants.exp_upper_bound | ErdosProblems | FormalConjectures/ErdosProblems/1063.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The least common multiple bound implies $n_k \le \exp((1 + o(1))k)$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1064 : {n | φ n > φ (n - φ n)}.HasDensity 1 | by
sorry | theorem | Erdos1064.erdos_1064 | ErdosProblems | FormalConjectures/ErdosProblems/1064.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$
have asymptotic density 1.
Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H
os concerning the arithmetical functions {$\phi$} and
{$\sigma$}. Colloq. Math. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1064.variants.k2 : {n | φ n < φ (n - φ n)}.Infinite | by
sorry | theorem | Erdos1064.erdos_1064.variants.k2 | ErdosProblems | FormalConjectures/ErdosProblems/1064.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$
such that $ϕ(n)< ϕ(n - ϕ(n))$
Reference: [GLW01] Grytczuk, A. and Luca, F. and W\'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the
{E}uler totient function. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1064.variants.general_function (f : ℕ → ℕ)
(hf : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))) :
{n : ℕ | φ (n - φ n) + f n < φ n}.HasDensity 1 | by
sorry | theorem | Erdos1064.erdos_1064.variants.general_function | ErdosProblems | FormalConjectures/ErdosProblems/1064.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For any function $f(n)=o(n)$,
we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$.
Reference:
[LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H
os concerning the arithmetical functions {$\phi$} and
{$\sigma$}. Colloq. Math. (2002), 111--130. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1065.parts.i :
answer(sorry) ↔ Set.Infinite {p | ∃ q k, p.Prime ∧ q.Prime ∧ p = 2^k * q + 1} | by
sorry | theorem | Erdos1065.erdos_1065.parts.i | ErdosProblems | FormalConjectures/ErdosProblems/1065.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Are there infinitely many primes $p$ such that $p = 2^k * q + 1$
for some prime $q$ and $k ≥ 0$?
This is mentioned as B46
in [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0)
by *Richard K. Guy* | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1065.parts.ii : answer(sorry) ↔
Set.Infinite {p | ∃ q k l, p.Prime ∧ q.Prime ∧ p = 2^k * 3^l * q + 1} | by
sorry | theorem | Erdos1065.erdos_1065.parts.ii | ErdosProblems | FormalConjectures/ErdosProblems/1065.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$
for some prime $q$ and $k ≥ 0$, $l ≥ 0$? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
InfinitelyEdgeConnected {V : Type*} (G : SimpleGraph V) : Prop | ∀ ⦃s : Set (Sym2 V)⦄, s.Finite → (G.deleteEdges s).Connected | def | Erdos1067.InfinitelyEdgeConnected | ErdosProblems | FormalConjectures/ErdosProblems/1067.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | A graph is infinitely edge-connected if to disconnect the graph requires deleting
infinitely many edges. In other words, removing any finite set of edges leaves
the graph connected. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1067 :
answer(False) ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
∃ (H : G.Subgraph), H.coe.chromaticCardinal = ℵ_ 1 ∧ InfinitelyConnected H.coe | by
sorry | theorem | Erdos1067.erdos_1067 | ErdosProblems | FormalConjectures/ErdosProblems/1067.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Does every graph with chromatic number $\aleph_1$ contain an infinitely connected subgraph with
chromatic number $\aleph_1$?
Komjáth [Ko13] proved that it is consistent that the answer is no. This was improved by
Soukup [So15], who constructed a counterexample using no extra set-theoretical assumptions. A
simpler elem... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1067.variants.infinite_edge_connectivity :
answer(False) ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
∃ (H : G.Subgraph), H.coe.chromaticCardinal = ℵ_ 1 ∧ InfinitelyEdgeConnected H.coe | by
sorry | theorem | Erdos1067.erdos_1067.variants.infinite_edge_connectivity | ErdosProblems | FormalConjectures/ErdosProblems/1067.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Thomassen [Th17] constructed a counterexample to the version which asks for infinite
edge-connectivity (that is, to disconnect the graph requires deleting infinitely many edges). | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1068 : answer(sorry) ↔
∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
∃ s : Set V, s.Countable ∧ InfinitelyConnected (G.induce s) | by
sorry | theorem | Erdos1068.erdos_1068 | ErdosProblems | FormalConjectures/ErdosProblems/1068.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is
infinitely connected? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
cardSet (n : ℕ) | { N | ∀ (pts : Finset ℝ²), pts.card = N → NonTrilinear (pts : Set ℝ²) →
HasConvexNGon n pts } | def | Erdos107.cardSet | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The set of $N$ such that any $N$ points in the plane, no three on a line,
contain a convex $n$-gon. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f (n : ℕ) : ℕ | sInf (cardSet n) | def | Erdos107.f | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The function $f(n)$ specified in `erdos_107`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_107 : answer(sorry) ↔ ∀ n ≥ 3, f n = 2^(n - 2) + 1 | by
sorry | theorem | Erdos107.erdos_107 | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line,
contain $n$ points which form the vertices of a convex $n$-gon.
Prove that $f(n) = 2^{n-2} + 1$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
nonempty_cardSet : ∀ n ≥ 3, (cardSet n).Nonempty | by
sorry | theorem | Erdos107.nonempty_cardSet | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line,
contain a convex $n$-gon. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f_zero_eq : f 0 = 0 | by
have : ∀ P, HasConvexNGon 0 P := by
intro; use ∅; simp [ConvexIndep]
simp [f, cardSet, this] | theorem | Erdos107.f_zero_eq | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Depending on details of definitions,
the statement is false or trivial for $n < 3$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f_three_eq : f 3 = 3 | by
sorry | theorem | Erdos107.f_three_eq | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
ersz_bounds :
∀ n ≥ 3, 2^(n - 2) + 1 ≤ f n ∧ f n ≤ Nat.choose (2 * n - 4) (n - 2) + 1 | by
sorry | theorem | Erdos107.variants.ersz_bounds | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős and Szekeres proved the bounds
$$
2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1
$$
([ErSz60] and [ErSz35] respectively).
[ErSz60] Erdős, P. and Szekeres, G., _On some extremum problems in elementary geometry_.
Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1960/61), 53-62.
[ErSz35] Erdős, P. and Szekeres, G., _... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
su_bound :
∃ r : ℕ → ℝ, r =o[atTop] (fun n => (n : ℝ)) ∧
∀ n ≥ 3, (f n : ℝ) ≤ 2^(n + r n) | by
sorry | theorem | Erdos107.variants.su_bound | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Suk [Su17] proved
$$
f(n) ≤ 2^{(1+o(1))n}.
$$
[Su17] Suk, Andrew, _On the Erdős-Szekeres convex polygon problem_.
J. Amer. Math. Soc. (2017), 1047-1053. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
hmpt_bound :
∃ r : ℕ → ℝ, r =O[atTop] (fun n => Real.sqrt (n * Real.log n)) ∧
∀ n ≥ 3, (f n : ℝ) ≤ 2^(n + r n) | by
sorry | theorem | Erdos107.variants.hmpt_bound | ErdosProblems | FormalConjectures/ErdosProblems/107.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20],
who prove
$$
f(n) ≤ 2^{n+O(\sqrt{n\log n})}.
$$
[HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor,
_Two extensions of the Erdős-Szekeres problem_. J. Eur. Math. Soc. (JEMS) (2020), 3981-3995. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
SegmentsDisjoint (seg1 seg2 : ℝ² × ℝ²) : Prop | segment ℝ seg1.1 seg1.2 ∩ segment ℝ seg2.1 seg2.2 ⊆ {seg1.1, seg1.2, seg2.1, seg2.2} | def | Erdos1071.SegmentsDisjoint | ErdosProblems | FormalConjectures/ErdosProblems/1071.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Two segments are disjoint if they only intersect at their endpoints (if at all). | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1071.parts.i :
answer(True) ↔ ∃ S : Finset (ℝ² × ℝ²),
Maximal (fun T : Finset (ℝ² × ℝ²) =>
(∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧
seg.1 0 ∈ Icc 0 1 ∧ seg.1 1 ∈ Icc 0 1 ∧
seg.2 0 ∈ Icc 0 1 ∧ seg.2 1 ∈ Icc 0 1) ∧
(T : Set (ℝ² × ℝ²)).Pairwise SegmentsDisjoint) S | by
sorry | theorem | Erdos1071.erdos_1071.parts.i | ErdosProblems | FormalConjectures/ErdosProblems/1071.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Can a finite set of disjoint unit segments in a unit square be maximal?
Solved affirmatively by [Da85], who gave an explicit construction.
This was formalized in Lean by Alexeev using Aristotle and ChatGPT. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1071.parts.ii :
answer(sorry) ↔ ∃ (R : Set ℝ²) (S : Set (ℝ² × ℝ²)),
IsOpen R ∧ IsConnected R ∧ S.Countable ∧ S.Infinite ∧
Maximal (fun T : Set (ℝ² × ℝ²) =>
(∀ seg ∈ T, dist seg.1 seg.2 = 1 ∧ seg.1 ∈ R ∧ seg.2 ∈ R) ∧
T.Pairwise SegmentsDisjoint) S | by
sorry | theorem | Erdos1071.erdos_1071.parts.ii | ErdosProblems | FormalConjectures/ErdosProblems/1071.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?
Solved affirmatively by [Fo99], who gave an explicit construction.
This was formalized in Lean by Alexeev using Aristotle and ChatGPT. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f (p : ℕ) : ℕ | sInf {n | (n)! + 1 ≡ 0 [MOD p]} | def | Erdos1072.f | ErdosProblems | FormalConjectures/ErdosProblems/1072.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For any prime $p$, let $f(p)$ be the least integer such that $f(p)! + 1 \equiv 0 \mod p$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1072.parts.i : answer(sorry) ↔ Set.Infinite {p | p.Prime ∧ f p = p - 1} | by
sorry | theorem | Erdos1072.erdos_1072.parts.i | ErdosProblems | FormalConjectures/ErdosProblems/1072.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is it true that there are infinitely many $p$ for which $f(p) = p − 1$? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1072.parts.ii :
answer(sorry) ↔ ∃ (P : Set ℕ), P ⊆ {p | p.Prime} ∧ P.HasDensity 1 {p | p.Prime} ∧
Tendsto (fun p => (f p / p : ℝ)) (atTop ⊓ principal P) (𝓝 0) | by
sorry | theorem | Erdos1072.erdos_1072.parts.ii | ErdosProblems | FormalConjectures/ErdosProblems/1072.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is it true that $f(p)/p \to 0$ for $p \to \infty$ in a density 1 subset of the primes? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1072.variants.littleo :
(fun x ↦ (({p | p.Prime ∧ f p = p - 1} ∩ Icc 0 x).ncard : ℝ)) =o[atTop]
(fun x ↦ x / Real.log x) | by
sorry | theorem | Erdos1072.erdos_1072.variants.littleo | ErdosProblems | FormalConjectures/ErdosProblems/1072.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \le x$ for which $f(p)=p−1$
is $o(x/\log x)$.
[HaSu02] Hardy, G. E. and Subbarao, M. V., _A modified problem of Pillai and some related questions._
Amer. Math. Monthly (2002), 554--559. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
A (x : ℕ) : ℝ | {u | u.Composite ∧ ∃ n, n ! + 1 ≡ 0 [MOD u] ∧ u < x}.ncard | def | Erdos1073.A | ErdosProblems | FormalConjectures/ErdosProblems/1073.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $A(x)$ count the number of composite $u < x$ such that $n!+1 \equiv 0 (\mod u)$ for some $n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1073 :
answer(sorry) ↔ ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧ ∀ x, A x ≤ x ^ (o x) | by
sorry | theorem | Erdos1073.erdos_1073 | ErdosProblems | FormalConjectures/ErdosProblems/1073.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is it true that $A(x) \le x^{o(1)}$? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
EHSNumbers : Set ℕ | {m | 1 ≤ m ∧ ∃ p, p.Prime ∧ ¬p ≡ 1 [MOD m] ∧ p ∣ m ! + 1} | abbrev | Erdos1074.EHSNumbers | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The EHS numbers (after Erdős, Hardy, and Subbarao) are those $m\geq 1$ such that there
exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
PillaiPrimes : Set ℕ | {p | p.Prime ∧ ∃ m ≥ 1, ¬p ≡ 1 [MOD m] ∧ p ∣ m ! + 1} | abbrev | Erdos1074.PillaiPrimes | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The Pillai primes are those primes $p$ such that there exists an $m \ge 1$ with
$p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
two_not_mem_pillaiPrimes : ¬ 2 ∈ PillaiPrimes | by
norm_num
intro m hm h
exact (Nat.dvd_factorial (by decide) (hm.lt_of_ne (by bound))).modEq_zero_nat.add_right 1 | theorem | Erdos1074.two_not_mem_pillaiPrimes | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
twentyThree_mem_pillaiPrimes : 23 ∈ PillaiPrimes | by
norm_num
use 14
decide | theorem | Erdos1074.twentyThree_mem_pillaiPrimes | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 | |
erdos_1074.parts.i : answer(sorry) ↔ ∃ c, EHSNumbers.HasDensity c | by
sorry | theorem | Erdos1074.erdos_1074.parts.i | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such
that $m! + 1 \equiv 0\pmod{p}$. Does
$$
\lim\frac{|S\cap[1, x]|}{x}
$$
exist? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.parts.ii : EHSNumbers.HasDensity answer(sorry) | by
sorry | theorem | Erdos1074.erdos_1074.parts.ii | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such
that $m! + 1 \equiv 0\pmod{p}$. What is
$$
\lim\frac{|S\cap[1, x]|}{x}?
$$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.parts.iii : answer(sorry) ↔ ∃ c, PillaiPrimes.HasDensity c {p | p.Prime} | by
sorry | theorem | Erdos1074.erdos_1074.parts.iii | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with
$p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then does
$$
\lim\frac{|P\cap[1, x]|}{\pi(x)}
$$
exist? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.parts.iv :
PillaiPrimes.HasDensity answer(sorry) {p | p.Prime} | by
sorry | theorem | Erdos1074.erdos_1074.parts.iv | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with
$p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then what is
$$
\lim\frac{|P\cap[1, x]|}{\pi(x)}?
$$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.variants.mem_pillaiPrimes : 23 ∈ PillaiPrimes | by
norm_num
exact ⟨14, by decide⟩ | theorem | Erdos1074.erdos_1074.variants.mem_pillaiPrimes | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Pillai [Pi30] raised the question of whether there exist any primes in $P$. This was answered
by Chowla, who noted that, for example, $14! + 1 \equiv 18! + 1 \equiv 0 \pmod{23}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.variants.EHSNumbers_infinite : EHSNumbers.Infinite | by
sorry | theorem | Erdos1074.erdos_1074.variants.EHSNumbers_infinite | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős, Hardy, and Subbarao proved that $S$ is infinite.
Formal proof linked here provided by AlphaProof. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.variants.PillaiPrimes_infinite : PillaiPrimes.Infinite | by
sorry | theorem | Erdos1074.erdos_1074.variants.PillaiPrimes_infinite | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős, Hardy, and Subbarao proved that $P$ is infinite. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.variants.EHSNumbers_init :
nth EHSNumbers '' (Set.Icc 0 6) = {8, 9, 13, 14, 15, 16, 17} | by
sorry | theorem | Erdos1074.erdos_1074.variants.EHSNumbers_init | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The sequence $S$ begins $8, 9, 13, 14, 15, 16, 17, ...$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.variants.PillaiPrimes_init :
nth PillaiPrimes '' (Set.Icc 0 5) = {23, 29, 59, 61, 67, 71} | by
sorry | theorem | Erdos1074.erdos_1074.variants.PillaiPrimes_init | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | The sequence $P$ begins $23, 29, 59, 61, 67, 71, ...$ | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1074.variants.EHSNumbers_one_half : EHSNumbers.HasDensity (1 / 2) | by
sorry | theorem | Erdos1074.erdos_1074.variants.EHSNumbers_one_half | ErdosProblems | FormalConjectures/ErdosProblems/1074.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Regarding the first question, Hardy and Subbarao computed all EHS numbers up to $2^{10}$, and
write "...if this trend conditions we expect [the limit] to be around 0.5, if it exists." | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1077 :
answer(False) ↔ ∀ ε > (0 : ℝ), ε < 1 → ∀ α > (0 : ℝ), α < 1 → ∀ᶠ D in atTop, ∀ᶠ n in atTop,
∀ G : SimpleGraph (Fin n), G.edgeSet.ncard > (n : ℝ) ^ (1 + α) →
∃ (H : Subgraph G),
letI m | H.verts.ncard
IsBalanced H.coe D ∧
m > (n : ℝ) ^ (1 - α) ∧
H.edgeSet.ncard > ε * m ^ (1 + α) := by
sorry | theorem | Erdos1077.erdos_1077 | ErdosProblems | FormalConjectures/ErdosProblems/1077.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | We call a graph $D$-balanced (or $D$-almost-regular) if the maximum degree is at most $D$ times the
minimum degree.
Let $ε, α > 0$ and $D$ and $n$ be sufficiently large. If $G$ is a graph on $n$ vertices with at
least $n^{1+α}$ edges, then must $G$ contain a $D$-balanced subgraph on $m > n^{1-α}$ vertices with
at leas... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_108 :
answer(sorry) ↔ ∀ r ≥ 4, ∀ k ≥ (2 : ℕ), ∃ (f : ℕ),
∀ (V : Type u) (G : SimpleGraph V) (_ : Nonempty V)
(hchro : f ≤ SimpleGraph.chromaticNumber G),
∃ (H : G.Subgraph), (SimpleGraph.girth H.coe ≥ r) ∧
(SimpleGraph.chromaticNumber H.coe ≥ k) | by
sorry | theorem | Erdos108.erdos_108 | ErdosProblems | FormalConjectures/ErdosProblems/108.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r)
contains a subgraph of girth ≥ r and chromatic number ≥ k? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
IsBipartition {V : Type*} (G : SimpleGraph V) (X Y : Set V) : Prop | Disjoint X Y ∧ X ∪ Y = Set.univ ∧ ∀ ⦃u v⦄, G.Adj u v → (u ∈ X ↔ v ∈ Y) | def | Erdos1080.IsBipartition | ErdosProblems | FormalConjectures/ErdosProblems/1080.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | `IsBipartition G X Y` means that `X` and `Y` form a bipartition of the vertices of `G`. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1080 :
answer(False) ↔
∃ c > (0 : ℝ), ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V),
IsBipartition G X Y → X.ncard = ⌊(Fintype.card V : ℝ) ^ (2/3 : ℝ)⌋₊ →
G.edgeSet.ncard ≥ c * Fintype.card V →
∃ (v : V) (walk : G.Walk v v), walk.IsCycle ∧ walk.length = 6 | by
sorry | theorem | Erdos1080.erdos_1080 | ErdosProblems | FormalConjectures/ErdosProblems/1080.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $G$ be a bipartite graph on $n$ vertices such that one part has $\lfloor n^{2/3}\rfloor$
vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must
contain a $C_6$?
The answer is no, as shown by De Caen and Székely [DeSz92], who in fact show a stronger result.
Let $f(n,m)$ be the ma... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1082.parts.i : answer(sorry) ↔ ∀ (A : Finset ℝ²) (hA_n3c : NonTrilinear (A : Set ℝ²)),
A.card / 2 ≤ distinctDistances A | by
sorry | theorem | Erdos1082.erdos_1082.parts.i | ErdosProblems | FormalConjectures/ErdosProblems/1082.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line.
Does $A$ determine at least $\lfloor n/2\rfloor$ distinct distances? | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1082.parts.ii : answer(False) ↔
∀ (A : Finset ℝ²) (hA : A.Nonempty) (hA_n3c : NonTrilinear (A : Set ℝ²)),
∃ (a : ℝ²) (ha : a ∈ A), A.card / 2 ≤ distinctDistancesFrom A a - 1 | by
sorry | theorem | Erdos1082.erdos_1082.parts.ii | ErdosProblems | FormalConjectures/ErdosProblems/1082.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line.
Must there exist a single point from which there are at least $\lfloor n/2\rfloor$ distinct
distances?
This question has been answered negatively by Xichuan in the
[comments](https://www.erdosproblems.com/forum/thread/1082), who gave a set of ... | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f (d n : ℕ) : ℕ | ⨆ (s : Finset (ℝ^ d)) (_ : s.card = n) (_ : IsSeparated' 1 (s : Set (ℝ^ d))), unitDistNum s | def | Erdos1084.f | ErdosProblems | FormalConjectures/ErdosProblems/1084.lean | [
"FormalConjectures.Util.ProblemImports"
] | [
"unitDistNum"
] | The maximal number of pairs of points which are distance 1 apart that a set of `n` 1-separated
points in `ℝ^d` make. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1084.variants.upper_d1 : f 1 n = n - 1 | by
sorry | theorem | Erdos1084.erdos_1084.variants.upper_d1 | ErdosProblems | FormalConjectures/ErdosProblems/1084.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | It is easy to check that $f_1(n) = n - 1$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1084.variants.easy_upper_d2 (hn : n ≠ 0) : f 2 n < 3 * n | by
sorry | theorem | Erdos1084.erdos_1084.variants.easy_upper_d2 | ErdosProblems | FormalConjectures/ErdosProblems/1084.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | It is easy to check that $f_2(n) < 3n$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1084.variants.upper_d2 : ∃ c > (0 : ℝ), ∀ n > 0, f 2 n < 3 * n - c * sqrt n | by
sorry | theorem | Erdos1084.erdos_1084.variants.upper_d2 | ErdosProblems | FormalConjectures/ErdosProblems/1084.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős showed that there is some constant $c > 0$ such that $f_2(n) < 3n - c n^{1/2}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1084.variants.triangular_optimal_d2 : f 2 (3 * n ^ 2 + 3 * n + 1) = 9 * n ^ 2 + 3 * n | by
sorry | theorem | Erdos1084.erdos_1084.variants.triangular_optimal_d2 | ErdosProblems | FormalConjectures/ErdosProblems/1084.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős conjectured that the triangular lattice is best possible in 2D, in particular that
$f_2(3n^2 + 3n + 1) < 9n^2 + 3n$.
Note: in [Er75f] is read $9n^2 + 6n$, but this seems to be a typo. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1084.variants.upper_lower_d3 :
∃ c₁ : ℝ, ∃ c₂ > (0 : ℝ), ∀ᶠ n in atTop,
6 * n - c₁ * n ^ (2 / 3 : ℝ) ≤ f 3 n ∧ f 3 n ≤ 6 * n - c₂ * n ^ (2 / 3 : ℝ) | by
sorry | theorem | Erdos1084.erdos_1084.variants.upper_lower_d3 | ErdosProblems | FormalConjectures/ErdosProblems/1084.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős claims the existence of two constants $c_1, c_2 > 0$
such that $6n - c_1 n^{2/3} ≤ f_3(n) \le 6n - c_2 n^{2/3}$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f (d n : ℕ) : ℕ | ⨆ (s : Finset (ℝ^ d)) (_ : s.card = n), unitDistNum s | def | Erdos1085.f | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [
"unitDistNum"
] | The maximal number of pairs of points which are distance 1 apart that a set of `n` points in
`ℝ^d` make. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1085.variants.lower_d2 :
∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, (n : ℝ) ^ (1 + c / log (log n)) < f 2 n | by
sorry | theorem | Erdos1085.erdos_1085.variants.lower_d2 | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős showed $f_2(n) > n^{1+c/\log\log n}$ for some $c > 0$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1085.variants.upper_d2 : (fun n ↦ (f 2 n : ℝ)) =O[atTop] (fun n ↦ (n : ℝ) ^ (4/3 : ℝ)) | by
sorry | theorem | Erdos1085.erdos_1085.variants.upper_d2 | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1085.variants.lower_d3 :
(fun n : ℕ ↦ (n : ℝ) ^ (4/3 : ℝ) * log (log n)) =O[atTop] (fun n ↦ (f 3 n : ℝ)) | by
sorry | theorem | Erdos1085.erdos_1085.variants.lower_d3 | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős showed $f_3(n) = Ω(n^{4/3}\log\log n)$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1085.variants.upper_d3 : answer(sorry) ↔
(fun n ↦ (f 3 n : ℝ)) =O[atTop] (fun n : ℕ ↦ (n : ℝ) ^ (4/3 : ℝ) * log (log n)) | by
sorry | theorem | Erdos1085.erdos_1085.variants.upper_d3 | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound?. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1085.variants.lower_d4_lenz (hd : 4 ≤ d) :
∃ C : ℝ, ∀ n : ℕ, ↑(d / 2 - 1) / (2 * ↑(d / 2)) * n ^ 2 - C ≤ f d n | by
sorry | theorem | Erdos1085.erdos_1085.variants.lower_d4_lenz | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Lenz showed that, for $d \ge 4$, $f_d(n) \ge \frac{p - 1}{2p} n^2 - O(1)$ where
$p = \lfloor\frac d2\rfloor$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1085.variants.upper_d4_erdos (hd : 4 ≤ d) :
∃ g : ℕ → ℝ, Tendsto g atTop (𝓝 0) ∧
∀ n, f d n ≤ (↑(d / 2 - 1) / (2 * ↑(d / 2)) + g n) * n ^ 2 | by
sorry | theorem | Erdos1085.erdos_1085.variants.upper_d4_erdos | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős showed that, for $d \ge 4$, $f_d(n) \le \left(\frac{p - 1}{2p} + o(1)\right) n^2$ where
$p = \lfloor\frac d2\rfloor$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_1085.variants.upper_lower_d5_odd (hd : 5 ≤ d) (hd_odd : Odd d) :
∃ c₁ > (0 : ℝ), ∃ c₂ : ℝ, ∀ᶠ n in atTop,
↑(d / 2 - 1) / (2 * ↑(d / 2)) * n ^ 2 + c₁ * n ^ (4 / 3 : ℝ) ≤ f d n ∧
f d n ≤ ↑(d / 2 - 1) / ↑(d / 2) * n ^ 2 + c₂ * n ^ (4 / 3 : ℝ) | by
sorry | theorem | Erdos1085.erdos_1085.variants.upper_lower_d5_odd | ErdosProblems | FormalConjectures/ErdosProblems/1085.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Erdős and Pach showed that, for $d \ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$
such that $\frac{p - 1}{2p} n^2 - c_1 n^{4/3} ≤ f_d(n) \le \frac{p - 1}{2p} n^2 + c_2 n^{4/3}$ where
$p = \lfloor\frac d2\rfloor$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
erdos_109 (A : Set ℕ) (h : A.upperDensity > 0) :
∃ B C : Set ℕ, B.Infinite ∧ C.Infinite ∧ B + C ⊆ A | by
sorry | theorem | Erdos109.erdos_109 | ErdosProblems | FormalConjectures/ErdosProblems/109.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | Any $A\subseteq \mathbb{N}$ of positive upper density contains a sumset $B+C$ where both $B$ and $C$
are infinite.
The Erdős sumset conjecture. Proved by Moreira, Richter, and Robertson [MRR19]. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
f (r n : ℕ) : ℕ | sSup {k : ℕ |
∀ G : SimpleGraph (Fin n),
(∀ H : Subgraph G,
∃ E : Finset (Sym2 H.verts),
E.card ≤ k ∧
chromaticNumber (H.coe.deleteEdges E) ≤ (r : ℕ∞)) →
chromaticNumber G ≤ (r + 1 : ℕ∞)} | def | Erdos1092.f | ErdosProblems | FormalConjectures/ErdosProblems/1092.lean | [
"FormalConjectures.Util.ProblemImports"
] | [] | $f_r(n)$ is maximal such that, if a graph $G$ on $n$ vertices has the property that every
subgraph $H$ on $m$ vertices has chromatic number $\leq r$ once we remove $f_r(m)$ edges
from it, then $G$ has chromatic number $\leq r+1$. | https://github.com/google-deepmind/formal-conjectures | b9b8aa0fd6170e482798f7c78c163acacd40e083 |
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