951.lean`

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

4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 951

References:

erdosproblems.com/951
[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos951.Erdos951Prop Documented declaration: Erdos951Prop line 39 source
Source documentation
A sequence a : ℕ → ℝ is said to have property Erdos951Prop if for any pair of distinct finitely supported sequences k l : ℕ →₀ ℕ their corresponding Beurling integers are of distance at least one apart.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Finsupp, Ne, GE.ge, abs, HSub.hSub, beurlingInteger, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.
```
def Erdos951Prop (a : ℕ → ℝ) : Prop :=
…
```
theorem Erdos951.erdos_951.variants.isBeurlingPrimes If a strictly increasing sequence a : ℕ → ℝ has property Erdos951Prop and 1 < a 0, then a is a sequence of Beurling prime numbers line 45 source
Source documentation
If a strictly increasing sequence a : ℕ → ℝ has property Erdos951Prop and 1 < a 0, then a is a sequence of Beurling prime numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LT.lt, OfNat.ofNat, StrictMono, Erdos951.Erdos951Prop, IsBeurlingPrimes
Proof uses: And.intro, And, Filter.Tendsto, Filter.atTop, Iff.mpr, Exists, LE.le, Filter.tendsto_atTop_atTop, Classical.byContradiction, Not, nhds, False, Exists.intro, iSup, tendsto_atTop_isLUB, StrictMono.monotone, isLUB_ciSup, Membership.mem, Set, upperBounds and 130 more
165 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_951.variants.isBeurlingPrimes {a : ℕ → ℝ} (ha : 1 < a 0)
    (hm : StrictMono a) (he : Erdos951Prop a) :
    IsBeurlingPrimes a := by
…
```
Mentions: Erdos951.Erdos951Prop, IsBeurlingPrimes
theorem Erdos951.erdos_951 If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x? sorry line 62 source
Source documentation
If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?

depends on sorry what this means
Statement uses: Iff, True, Nat, Real, LT.lt, OfNat.ofNat, StrictMono, Erdos951.Erdos951Prop, Filter.Eventually, LE.le, Set.ncard, Set.ofPred, Nat.primeCounting, Nat.floor, Filter.atTop
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_951 : answer(sorry) ↔
    ∀ a : ℕ → ℝ, 1 < a 0 → StrictMono a → Erdos951Prop a →
      ∀ᶠ (x : ℝ) in Filter.atTop, {i : ℕ | a i ≤ x}.ncard ≤ π ⌊x⌋₊ := by
…
```
Mentions: Erdos951.Erdos951Prop
theorem Erdos951.erdos_951.variants.beurling Documented declaration: beurling sorry line 72 source
Source documentation
Beurling conjectured that if 1 < a 0 < a 1 < ⋯ has property Erdos951Prop and the number of reals in $[1, x]$ of the form $\prod_i a_i^{k_i}$ is $x + o(\log x)$, then a must be the sequence of primes. Property Erdos951Prop makes distinct exponent vectors give distinct Beurling integers, so counting the set BeurlingIntegers a counts the generalised integers.

depends on sorry what this means
Statement uses: Nat, Real, IsBeurlingPrimes, Erdos951.Erdos951Prop, Asymptotics.IsLittleO, Filter.atTop, HSub.hSub, Set.ncard, Inter.inter, Set, BeurlingIntegers, Set.Iic, Real.log, Eq, Function.comp, Nat.nth, Nat.Prime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_951.variants.beurling :
    ∀ a : ℕ → ℝ, IsBeurlingPrimes a → Erdos951Prop a →
    ((fun x => (BeurlingIntegers a ∩ .Iic x).ncard - x) =o[atTop] Real.log) →
    a = Nat.cast ∘ Nat.nth Nat.Prime := by
…
```
Mentions: IsBeurlingPrimes, Erdos951.Erdos951Prop, BeurlingIntegers