FormalConjectures.ErdosProblems.«478»

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module

public import FormalConjecturesUtil

Erdős Problem 478

References:

erdosproblems.com/478

[AnTa16] V. Andrejić and M. Tatarevic, On distinct residues of factorials. arXiv:1603.04086 (2016).

[GSSV24] Grebennikov, Alexandr and Sagdeev, Arsenii and Semchankau, Aliaksei and Vasilevskii, Aliaksei, On the sequence {$n! \bmod p$}. Rev. Mat. Iberoam. (2024), 637--648.

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

[KlMu17] Klurman, Oleksiy and Munsch, Marc, Distribution of factorials modulo {$p$}. J. Théor. Nombres Bordeaux (2017), 169--177.

[RoSc60] Rokowska, B. and Schinzel, A., Sur un problème de {M}. {E}rdős. Elem. Math. (1960), 84--85.

[Tr13] T. Trudgian, There are no socialist primes less than $10^9$. arXiv:1310.6403 (2013).

@[expose] public sectionnamespace Erdos478

Let $p$ be a prime and $$A_p = { k! \pmod{p} : 1\leq k<p}.$$ Is it true that $$\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?$$

@[category research open, AMS 11]
theorem erdos_478 : answer(sorry) ↔
    Filter.Tendsto
      (fun p : ℕ =>
        (((Finset.Ico 1 p).image (fun k => Nat.factorial k % p)).card : ℝ) / p)
      (Filter.atTop ⊓ Filter.principal {p : ℕ | p.Prime})
      (nhds (1 - 1 / Real.exp 1)) := by⊢ True ↔
  Filter.Tendsto (fun p ↦ ↑(Finset.image (fun k ↦ k.factorial % p) (Finset.Ico 1 p)).card / ↑p)
    (Filter.atTop ⊓ Filter.principal {p | Nat.Prime p}) (nhds (1 - 1 / Real.exp 1))
  sorryAll goals completed! 🐙

end Erdos478