940.lean`

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

5 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 940

References:

erdosproblems.com/940
[BaBr94] Baker, R. C. and Brüdern, J., On sums of two squarefull numbers. Math. Proc. Cambridge Philos. Soc. (1994), 1-5.
[He88] Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. (1988), 137-163.

Imports (1)

FormalConjecturesUtil

theorem Erdos940.erdos_940 Let $r \ge 3$ sorry line 42 source
Source documentation
Let $r \ge 3$. Is it true that the set of integers which are the sum of at most $r$ $r$-powerful numbers has density $0$?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Set.HasDensity, Set.ofPred, Exists, Multiset, And, LE.le, Multiset.card, Membership.mem, Nat.Full, Eq, Multiset.sum, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_940 :
    answer(sorry) ↔ ∀ r ≥ 3,
      {n : ℕ | ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ n = S.sum}.HasDensity 0 := by
…
```
theorem Erdos940.erdos_940.variants.two The set of integers which are the sum of at most two $2$-powerful numbers has density $0$ sorry line 53 source
Source documentation
The set of integers which are the sum of at most two $2$-powerful numbers has density $0$.
Erdős called this 'easy'. Baker and Brüdern [BaBr94] gave the first proof in the literature.

depends on sorry what this means
Statement uses: Set.HasDensity, Nat, Set.ofPred, Exists, Multiset, And, LE.le, Multiset.card, OfNat.ofNat, Membership.mem, Nat.Full, Eq, Multiset.sum, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_940.variants.two :
    {n : ℕ | ∃ (S : Multiset ℕ),
      S.card ≤ 2 ∧ (∀ s ∈ S, (2).Full s) ∧ n = S.sum}.HasDensity 0 := by
…
```
theorem Erdos940.erdos_940.variants.three_cubes Is it true that the set of integers which are the sum of at most three cubes has density $0$? sorry line 66 source
Source documentation
Is it true that the set of integers which are the sum of at most three cubes has density $0$?
The cubes are those of non-negative integers, which is what Multiset ℕ gives. This choice decides the question: over ℤ a sum of three cubes is conjectured to represent every integer that is not $\pm 4 \bmod 9$, which is density $7/9$.

depends on sorry what this means
Statement uses: Iff, True, Set.HasDensity, Nat, Set.ofPred, Exists, Multiset, And, LE.le, Multiset.card, OfNat.ofNat, Eq, Multiset.sum, Multiset.map, HPow.hPow, Real, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_940.variants.three_cubes :
    answer(sorry) ↔
    {n : ℕ | ∃ (S : Multiset ℕ), S.card ≤ 3 ∧ n = (Multiset.map (· ^ 3) S).sum}.HasDensity 0 := by
…
```
theorem Erdos940.erdos_940.variants.large_integers Let $r \ge 3$ sorry line 77 source
Source documentation
Let $r \ge 3$. Are there infinitely many integers which are not the sum of at most $r$-many $r$-powerful numbers?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Not, Filter.Eventually, Exists, Multiset, And, LE.le, Multiset.card, Membership.mem, Nat.Full, Eq, Multiset.sum, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_940.variants.large_integers :
    answer(sorry) ↔
    ∀ r ≥ 3, ¬ (∀ᶠ x in atTop, ∃ (S : Multiset ℕ), S.card ≤ r ∧ (∀ s ∈ S, r.Full s) ∧ x = S.sum) := by
…
```
theorem Erdos940.erdos_940.variants.three_powerful Heath-Brown [He88] has proved that all large numbers are the sum of at most three $2$-powerful numbers sorry line 90 source
Source documentation
Heath-Brown [He88] has proved that all large numbers are the sum of at most three $2$-powerful numbers.
This is the case $r = 2$ of Erdos1107.erdos_1107, which asks the same question with $r + 1$ summands, and it is stated there as Erdos1107.erdos_1107.variants.two.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Exists, Multiset, And, LE.le, Multiset.card, OfNat.ofNat, Membership.mem, Nat.Full, Eq, Multiset.sum, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_940.variants.three_powerful :
    ∀ᶠ x in atTop, ∃ (S : Multiset ℕ), S.card ≤ 3 ∧ (∀ s ∈ S, (2).Full s) ∧ x = S.sum := by
…
```