FormalConjectures.ErdosProblems.«10»

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import FormalConjecturesUtil

Erdős Problem 10

Reference: erdosproblems.com/10


namespace Erdos10

The set of natural numbers that can be written as a sum of a prime and at most $k$ powers of $2$.

abbrev sumPrimeAndTwoPows (k : ℕ) : Set ℕ :=
  { p + (pows.map (2 ^ ·)).sum | (p : ℕ) (pows : Multiset ℕ) (_ : p.Prime)
    (_ : pows.card ≤ k)}

Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$?

@[category research open, AMS 5 11]
theorem declaration uses 'sorry'erdos_10 : answer(sorry) ↔ ∃ k, sumPrimeAndTwoPows k = Set.univ \ {0, 1} := by⊢ True ↔ ∃ k, sumPrimeAndTwoPows k = Set.univ \ {0, 1}
  sorryAll goals completed! 🐙

Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$ such that the set of integers which are the sum of a prime and at most $k(ϵ)$ many powers of $2$ has lower density at least $1 - ϵ$.

Ref: Gallagher, P. X., Primes and powers of 2.

@[category research solved, AMS 5 11]
theorem declaration uses 'sorry'erdos_10.variants.gallagher (ε : ℝ)
    (hε : 0 < ε) : ∃ k, 1 - ε ≤ (sumPrimeAndTwoPows k).lowerDensity := byε:ℝhε:0 < ε⊢ ∃ k, 1 - ε ≤ (sumPrimeAndTwoPows k).lowerDensity
  sorryAll goals completed! 🐙

Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers.

Ref: Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$

@[category research open, AMS 5 11]
theorem declaration uses 'sorry'erdos_10.variants.granville_soundararajan_odd :
    {n : ℕ | Odd n ∧ 1 < n} ⊆ sumPrimeAndTwoPows 3 ∧
      {n : ℕ | Even n ∧ n ≠ 0} ⊆ sumPrimeAndTwoPows 4 := by⊢ {n | Odd n ∧ 1 < n} ⊆ sumPrimeAndTwoPows 3 ∧ {n | Even n ∧ n ≠ 0} ⊆ sumPrimeAndTwoPows 4
  sorryAll goals completed! 🐙

Bogdan Grechuk has observed that 1117175146 is not the sum of a prime and at most $3$ powers of $2$.

@[category research solved, AMS 5 11]
theorem declaration uses 'sorry'erdos_10.variants.grechuk_example :
    1117175146 ∉ sumPrimeAndTwoPows 3 := by⊢ 1117175146 ∉ sumPrimeAndTwoPows 3
  sorryAll goals completed! 🐙

There are infinitely many even integers not the sum of a prime and $2$ powers of $2$

@[category research solved, AMS 5 11]
theorem declaration uses 'sorry'erdos_10.variants.two_pows :
    Set.Infinite <| {n : ℕ | Even n} \ sumPrimeAndTwoPows 2 := by⊢ ({n | Even n} \ sumPrimeAndTwoPows 2).Infinite
  sorryAll goals completed! 🐙

Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).

@[category research open, AMS 5 11]
theorem declaration uses 'sorry'erdos_10.variants.grechuk :
    Set.Infinite <| {n : ℕ | Even n} \ sumPrimeAndTwoPows 3 := by⊢ ({n | Even n} \ sumPrimeAndTwoPows 3).Infinite
  sorryAll goals completed! 🐙


end Erdos10