1210.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1210

References:

erdosproblems.com/1210
[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Imports (1)

FormalConjecturesUtil

theorem Erdos1210.erdos_1210 Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$ sorry line 42 source
Source documentation
Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$. Is it true that $\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p < n}\frac{1}{p}+O(1)$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, Nat, Finset, Membership.mem, And, LE.le, OfNat.ofNat, LT.lt, Ne, Nat.Coprime, Finset.sum, HDiv.hDiv, HSub.hSub, HAdd.hAdd, Finset.filter, Prime, Finset.range
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1210 :
  answer(sorry) ↔
    ∃ C : ℝ, ∀ n : ℕ, ∀ A : Finset ℕ,
      (∀ a ∈ A, 1 ≤ a ∧ a < n) →
      (∀ a ∈ A, ∀ b ∈ A, a ≠ b → a.Coprime b) →
      ∑ a ∈ A, (1 / ((n : ℝ) - a)) ≤ (∑ p ∈ (range n).filter Prime, (1 / (p : ℝ))) + C := by
…
```
theorem Erdos1210.erdos_1210.variants.er80_correction In [Er80] he claims he "did not state this quite correctly" in [Er77c] sorry line 56 source
Source documentation
In [Er80] he claims he "did not state this quite correctly" in [Er77c]. The problem in [Er77c] which Erdős is presumably referring to states that if $n < q_1 < \cdots < q_k\leq m$ is the set of primes in $(n,m]$ then $\sum \frac{1}{q_i-n} < \sum_{p < m-n}\frac{1}{p}+O(1)$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, Nat, LT.lt, Finset.sum, Finset.filter, Prime, Finset.Ioc, HDiv.hDiv, OfNat.ofNat, HSub.hSub, HAdd.hAdd, Finset.range
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
39 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1210.variants.er80_correction :
  answer(sorry) ↔
    ∃ C : ℝ, ∀ n m : ℕ, n < m →
      ∑ q ∈ (Ioc n m).filter Prime, (1 / ((q : ℝ) - n)) <
      (∑ p ∈ (range (m - n)).filter Prime, (1 / (p : ℝ))) + C := by
…
```