FormalConjectures.ErdosProblems.«369»

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import FormalConjecturesUtil

Erdős Problem 369

References:

erdosproblems.com/369

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

[EgSe76] Eggleton, R. B. and Selfridge, J. L., Consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1976), 1--11.

[BFMW20] Bober, J. W. and Fretwell, D. and Martin, G. and Wooley, T. D., Smooth values of polynomials. J. Aust. Math. Soc. (2020), 245--261.

[BaWo98] Balog, Antal and Wooley, Trevor D., On strings of consecutive integers with no large prime factors. J. Austral. Math. Soc. Ser. A (1998), 266-276.

open Filternamespace Erdos369

Let $\epsilon>0$ and $k\geq 2$. Is it true that, for all sufficiently large $n$, there is a sequence of $k$ consecutive integers in ${1,\ldots,n}$ all of which are $n^\epsilon$-smooth?

The problem is trivially true as written (simply taking ${1,\ldots,k}$ and $n>k^{1/\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is not clear which Erdős and Graham meant. We formalize the second: each $m\in P$ (where $P$ is the sequence of $k$ consecutive integers sought for) must be in $[n/2,n]$. In this case a positive answer also follows directly from the result of Balog and Wooley [BaWo98] for infinitely many $n$. Proving this is true for all large $n$ does not follow immediately from [BaWo98], but can be deduced using a similar construction, as shown by SkyYang.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos369.lean"]
theorem erdos_369 : answer(True) ↔
    ∀ (ε : ℝ) (hε : 0 < ε) (k : ℕ) (hk : 2 ≤ k),
      ∀ᶠ (n : ℕ) in atTop, ∃ a : ℕ, n / 2 ≤ a + 1 ∧ a + k ≤ n ∧
        ∀ j < k, ∀ p ∈ (a + 1 + j).primeFactors, (p : ℝ) ≤ (n : ℝ) ^ ε := by⊢ True ↔
  ∀ (ε : ℝ),
    0 < ε →
      ∀ (k : ℕ),
        2 ≤ k →
          ∀ᶠ (n : ℕ) in atTop, ∃ a, n / 2 ≤ a + 1 ∧ a + k ≤ n ∧ ∀ j < k, ∀ p ∈ (a + 1 + j).primeFactors, ↑p ≤ ↑n ^ ε
  sorryAll goals completed! 🐙

end Erdos369