45.lean`

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L4DEL: Lean4 Domain Explanation Language

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1 declarations, 1 contain sorry, 0 declared as axioms

Ben Green's Open Problem 45

Can we pick residue classes $a_p \pmod{p}$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least 10 of them?

References:

The version of this problem with $10$ replaced by $2$ is Erdos689.erdos_689 in FormalConjectures.ErdosProblems.«689».

Imports (2)

theorem Green45.green_45 Can we pick residue classes $a_p \pmod{p}$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least $10$ of them? sorry line 47 source
Source documentation
Can we pick residue classes $a_p \pmod{p}$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least $10$ of them?
Erdős remarks that he does not know how to answer it with $10$ replaced by $2$; this is Erdos689.erdos_689.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Exists, Membership.mem, Finset, Finset.Icc, OfNat.ofNat, LE.le, Finset.card, Finset.filter, And, Nat.Prime, Nat.ModEq, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_45 :
    answer(sorry) ↔ ∀ᶠ N in .atTop, ∃ a : ℕ → ℕ, ∀ m ∈ Finset.Icc 1 N,
      10 ≤ (Finset.Icc 1 N |>.filter fun p => p.Prime ∧ a p ≡ m [MOD p]).card := by
…
```