646.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 646

References:

erdosproblems.com/646
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
[Be97] Berend, Daniel, On the parity of exponents in the factorization of $n!$. J. Number Theory (1997), 13-19.

Imports (1)

FormalConjecturesUtil

theorem Erdos646.erdos_646 Let $p_1,\ldots,p_k$ be distinct primes sorry line 46 source
Source documentation
Let $p_1,\ldots,p_k$ be distinct primes. Are there infinitely many $n$ such that $n!$ is divisible by an even power of each of the $p_i$?
The answer is yes, proved by Berend [Be97], who further proved that the sequence of such $n$ has bounded gaps (where the bound depends on the initial set of primes).

depends on sorry what this means
Statement uses: Iff, True, Finset, Nat, Membership.mem, Nat.Prime, Set.Infinite, Set.ofPred, Even, padicValNat, Nat.factorial
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_646 : answer(True) ↔
    ∀ S : Finset ℕ, (∀ p ∈ S, p.Prime) →
      {n : ℕ | ∀ p ∈ S, Even (padicValNat p (n !))}.Infinite := by
…
```