970.lean`

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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 970

References:

erdosproblems.com/970
[FGKMT18] Ford, Kevin and Green, Ben and Konyagin, Sergei and Maynard, James and Tao, Terence, Long gaps between primes. J. Amer. Math. Soc. (2018), 65-105.
[Iw78] Iwaniec, Henryk, On the problem of {J}acobsthal. Demonstratio Math. (1978), 225--231.

Imports (1)

FormalConjecturesUtil

def Erdos970.IsJacobsthalBound IsJacobsthalBound k m says that every interval of m consecutive integers contains an integer coprime to every positive natural number having at most k distinct prime factors line 38 source
Source documentation
IsJacobsthalBound k m says that every interval of m consecutive integers contains an integer coprime to every positive natural number having at most k distinct prime factors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: LT.lt, OfNat.ofNat, LE.le, Finset.card, Nat.primeFactors, Int, Exists, And, Nat.Coprime, Int.natAbs, HAdd.hAdd
7 instances, coercions and auxiliary constants left out.
```
def IsJacobsthalBound (k m : ℕ) : Prop :=
…
```
def Erdos970.jacobsthalFunction Jacobsthal's function, uniformly parametrized by the maximum number of distinct prime factors line 45 source
Source documentation
Jacobsthal's function, uniformly parametrized by the maximum number of distinct prime factors.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Erdos970.IsJacobsthalBound
1 instances, coercions and auxiliary constants left out.
```
noncomputable def jacobsthalFunction (k : ℕ) : ℕ :=
…
```
theorem Erdos970.erdos_970 Documented declaration: erdos_970 sorry line 52 source
Source documentation
Let $h(k)$ be Jacobsthal's function, defined to as the minimal $m$ such that, if $n$ has at most $k$ prime factors, then in any set of $m$ consecutive integers there exists an integer coprime to $n$. Determine the order of magnitude of $h(k)$. In particular, is it true that $$h(k) \ll k^2?$$

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Nat, LT.lt, LE.le, Erdos970.jacobsthalFunction, HMul.hMul, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_970 : answer(sorry) ↔
    ∃ C > (0 : ℝ), ∀ k : ℕ, 0 < k → (jacobsthalFunction k : ℝ) ≤ C * k ^ 2 := by
…
```
Mentions: Erdos970.jacobsthalFunction