437.lean`

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L4DEL: Lean4 Domain Explanation Language

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 437

References:

erdosproblems.com/437
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[BPZ24] Bui, Hung M. and Pratt, Kyle and Zaharescu, Alexandru, A problem of Erdős-Graham-Granville-Selfridge on integral points on hyperelliptic curves. Math. Proc. Cambridge Philos. Soc. (2024), 309--323.
[Ta24] Tao, Terence, A result of Bui–Pratt–Zaharescu, and Erdős problem #437. Blog post (2024).

Imports (1)

FormalConjecturesUtil

def Erdos437.squarePartialProducts The number of partial products $a_1, a_1a_2, \ldots, a_1\cdots a_k$ of a finite sequence a = [a₁, …, a_k] which are squares line 42 source
Source documentation
The number of partial products $a_1, a_1a_2, \ldots, a_1\cdots a_k$ of a finite sequence a = [a₁, …, a_k] which are squares.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat
Proof uses: Finset.card, Finset.filter, IsSquare, List.prod, List.take, HAdd.hAdd, OfNat.ofNat, Finset.range, List.length
6 instances, coercions and auxiliary constants left out.
```
def squarePartialProducts (a : List ℕ) : ℕ :=
…
```
def Erdos437.L L x is the maximal number of square partial products of a sequence $1\leq a_1<\cdots<a_k\leq x$ line 47 source
Source documentation
L x is the maximal number of square partial products of a sequence $1\leq a_1<\cdots<a_k\leq x$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, List, And, List.Pairwise, LT.lt, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, Eq, Erdos437.squarePartialProducts
8 instances, coercions and auxiliary constants left out.
```
noncomputable def L (x : ℕ) : ℕ :=
…
```
theorem Erdos437.erdos_437 sorry line 61 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, HPow.hPow, HSub.hSub, Erdos437.L, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos437.erdos_437.variants.little_o Erdős and Graham write it is 'trivial' that there are $o(x)$ many such squares, although this is not quite trivial, using Siegel's theorem sorry line 70 source
Source documentation
Erdős and Graham write it is 'trivial' that there are $o(x)$ many such squares, although this is not quite trivial, using Siegel's theorem.

depends on sorry what this means
Statement uses: Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos437.L
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_437.variants.little_o :
    (fun x : ℕ ↦ (L x : ℝ)) =o[atTop] fun x : ℕ ↦ (x : ℝ) := by
…
```
Mentions: Erdos437.L
theorem Erdos437.erdos_437.variants.tao Tao [Ta24] shows that, if $u(x)=(\log x\log\log x)^{1/2}$, then $$x\exp(-(2^{1/2}+o(1))u(x)) \leq L(x) \leq x\exp(-(2^{-1/2}+o(1))u(x)).$$ sorry line 79 source
Source documentation
Tao [Ta24] shows that, if $u(x)=(\log x\log\log x)^{1/2}$, then $$x\exp(-(2^{1/2}+o(1))u(x)) \leq L(x) \leq x\exp(-(2^{-1/2}+o(1))u(x)).$$

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Filter.Eventually, LE.le, HMul.hMul, Real.exp, Neg.neg, HAdd.hAdd, Real.sqrt, Real.log, Erdos437.L, Inv.inv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_437.variants.tao :
    ∃ o₁ o₂ : ℕ → ℝ, Tendsto o₁ atTop (nhds 0) ∧ Tendsto o₂ atTop (nhds 0) ∧
      ∀ᶠ x : ℕ in atTop,
        x * exp (-(√2 + o₁ x) * √(log x * log (log x))) ≤ L x ∧
        L x ≤ x * exp (-((√2)⁻¹ + o₂ x) * √(log x * log (log x))) := by
…
```
Mentions: Erdos437.L