HardyLittlewood.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

9 declarations, 3 contain sorry, 0 declared as axioms

First Hardy–Littlewood conjecture

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/ErdosProblems/855.lean

def HardyLittlewood.IsPrimeConstellation A prime constellation is a tuple $(p, p + m_1, \dots, p + m_k)$ such that the $m_i$ are all positive even integers and every entry is a prime number line 40 source
Source documentation
A prime constellation is a tuple $(p, p + m_1, \dots, p + m_k)$ such that the $m_i$ are all positive even integers and every entry is a prime number.

standard axioms: propext what this means
Statement uses: Nat, Fin, Nat.succ
Proof uses: And, Eq, OfNat.ofNat, Ne, LT.lt, Nat.Prime, HAdd.hAdd, HMul.hMul
8 instances, coercions and auxiliary constants left out.
```
def IsPrimeConstellation {k : ℕ} (m : Fin k.succ → ℕ) (p : ℕ) : Prop :=
…
```
def HardyLittlewood.IsAdmissiblePrimeConstellation A prime constellation is said to be admissible if its elements do not form a complete set of residue classes with respect to any prime line 47 source
Source documentation
A prime constellation is said to be admissible if its elements do not form a complete set of residue classes with respect to any prime.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Fin, Nat.succ
Proof uses: And, HardyLittlewood.IsPrimeConstellation, Nat.Prime, Not, Function.Surjective, ZMod, HAdd.hAdd, HMul.hMul, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.
```
def IsAdmissiblePrimeConstellation {k : ℕ} (m : Fin k.succ → ℕ) (p : ℕ) : Prop :=
…
```
def HardyLittlewood.Nat.numResidues The number of distinct residue classes amongst a tuple $(m_0, \dots, m_k)$ for a prime $q$ line 53 source
Source documentation
The number of distinct residue classes amongst a tuple $(m_0, \dots, m_k)$ for a prime $q$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Nat.succ
Proof uses: Set.ncard, ZMod, Set.range
6 instances, coercions and auxiliary constants left out.
```
noncomputable def Nat.numResidues (q : ℕ) {k : ℕ} (m : Fin k.succ → ℕ) : ℕ :=
…
```
def HardyLittlewood.Nat.primeTupleCounting For a given tuple $(m_1, \dots, m_k)$, this counts number of admissible prime constellations $(p, p + m_1, \dots, p + m_k)$ where $p \leq n$ line 60 source
Source documentation
For a given tuple $(m_1, \dots, m_k)$, this counts number of admissible prime constellations $(p, p + m_1, \dots, p + m_k)$ where $p \leq n$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Nat.succ
Proof uses: Nat.count, HardyLittlewood.IsAdmissiblePrimeConstellation, Classical.propDecidable
```
noncomputable def Nat.primeTupleCounting {k : ℕ} (m : Fin k.succ → ℕ) (n : ℕ) : ℕ :=
…
```
def HardyLittlewood.FirstHardyLittlewoodConjectureFor The first Hardy–Littlewood conjecture for the tuple line 71 source
Source documentation
The first Hardy–Littlewood conjecture for the tuple $(m_0, m_1, \dots, m_k)$: if $m_0 = 0$ and the $m_i$ are pairwise distinct, then $$ \pi_P(n)\sim C_P\int_2^n\frac{dt}{\log^{k+1}t}. $$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Nat.succ
Proof uses: Eq, OfNat.ofNat, Function.Injective, Real, HMul.hMul, HPow.hPow, tprod, Subtype, And, Nat.Prime, LE.le, HDiv.hDiv, HSub.hSub, HardyLittlewood.Nat.numResidues, SummationFilter.unconditional, HardyLittlewood.Nat.primeTupleCounting, Asymptotics.IsEquivalent, Filter.atTop, intervalIntegral, Real.log and 1 more
38 instances, coercions and auxiliary constants left out.
```
def FirstHardyLittlewoodConjectureFor {k : ℕ} (m : Fin k.succ → ℕ) : Prop :=
…
```
theorem HardyLittlewood.first_hardy_littlewood_conjecture Let $P = (m_1, \dots, m_k)$ be a tuple of distinct positive even integers sorry line 94 source
Source documentation
Let $P = (m_1, \dots, m_k)$ be a tuple of distinct positive even integers. Let $\pi_P(n)$ denote the number of primes $p\leq n$ such that $(p, p + m_1, \dots, p + m_k)$ forms an admissible prime constellation. Let $w(q; m_1, \dots, m_k)$ denote the number of distinct residues of $0, m_1, \dots, m_k$ modulo $q$, and let $$ C_P = 2 ^ k\prod_{\substack{q\ \text{prime} \\ q\geq 3}} \frac{1 - \frac{w(q; m_1, \dots, m_k)}{q}}{\left(1 - \frac{1}{q}\right)^{k+1}}. $$ Then $$ \pi_P(n)\sim C_P\int_2^n\frac{dt}{\log^{k+1}t}. $$

depends on sorry what this means
Statement uses: Nat, Fin, Nat.succ, HardyLittlewood.FirstHardyLittlewoodConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem first_hardy_littlewood_conjecture {k : ℕ} (m : Fin k.succ → ℕ) :
    FirstHardyLittlewoodConjectureFor m := by
…
```
Mentions: HardyLittlewood.FirstHardyLittlewoodConjectureFor
def HardyLittlewood.SecondHardyLittlewoodConjectureFor Wikipedia URL: https://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture ## Second Hardy-Littlewood Conjecture line 100 source
Source comment
Wikipedia URL: https://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture ## Second Hardy-Littlewood Conjecture

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: LE.le, Nat.primeCounting, HAdd.hAdd
3 instances, coercions and auxiliary constants left out.
```
def SecondHardyLittlewoodConjectureFor (x y : ℕ) : Prop :=
…
```
theorem HardyLittlewood.second_hardy_littlewood_conjecture For integers $x, y \geq 2$, $$ \pi(x + y) \leq \pi(x) + \pi(y), $$ where $\pi(z)$ denotes the prime-counting function, giving the number of primes up to and including $z$ sorry line 112 source
Source documentation
For integers $x, y \geq 2$, $$ \pi(x + y) \leq \pi(x) + \pi(y), $$ where $\pi(z)$ denotes the prime-counting function, giving the number of primes up to and including $z$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HardyLittlewood.SecondHardyLittlewoodConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem second_hardy_littlewood_conjecture {x y : ℕ} (hx : 2 ≤ x) (hy : 2 ≤ y) :
    SecondHardyLittlewoodConjectureFor x y := by
…
```
Mentions: HardyLittlewood.SecondHardyLittlewoodConjectureFor
theorem HardyLittlewood.not_first_and_secondHardyLittlewoodConjecture Richards [Ri74] showed that only one of the two Hardy-Littlewood conjectures can be true sorry line 121 source
Source documentation
Richards [Ri74] showed that only one of the two Hardy-Littlewood conjectures can be true.
[Ri74] Richards, Ian (1974). On the Incompatibility of Two Conjectures Concerning Primes. Bull. Amer. Math. Soc. 80: 419–438.

depends on sorry what this means
Statement uses: Nat, Fin, Nat.succ, HardyLittlewood.FirstHardyLittlewoodConjectureFor, Not, LE.le, OfNat.ofNat, HardyLittlewood.SecondHardyLittlewoodConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem not_first_and_secondHardyLittlewoodConjecture :
    (∀ {k : ℕ} (m : Fin k.succ → ℕ), FirstHardyLittlewoodConjectureFor m) →
      ¬(∀ {x y : ℕ} (hx : 2 ≤ x) (hy : 2 ≤ y), SecondHardyLittlewoodConjectureFor x y) := by
…
```
Mentions: HardyLittlewood.FirstHardyLittlewoodConjectureFor, HardyLittlewood.SecondHardyLittlewoodConjectureFor