FormalConjectures.ErdosProblems.«453»

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import FormalConjecturesUtil

Erdős Problem 453

References:

erdosproblems.com/453

[Er70b] Erdős, P., Some applications of graph theory to number theory. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (1970), 136-145.

[Er74b] Erdős, P., Remarks on some problems in number theory. Math. Balkanica (1974), 197-202.

[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.

[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

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

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

[Po79] Pomerance, Carl, The prime number graph. Math. Comp. (1979), 399-408.


namespace Erdos453

The eventual prime inequality asked in Erdős Problem 453, using Mathlib's zero-based primes.

def EventuallyHasPrimeWitness : Prop :=
  ∃ N : ℕ, ∀ n : ℕ, N ≤ n →
    ∃ i : ℕ, i < n ∧
      (Nat.nth Nat.Prime n) ^ (2 : ℕ) <
        Nat.nth Nat.Prime (n + i) * Nat.nth Nat.Prime (n - i)

Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that $$ p_n^2 < p_{n+i}p_{n-i}, $$ where $p_k$ is the $k$th prime?

Pomerance proved that the answer is no.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos453.lean"]
theorem declaration uses 'sorry'erdos_453 : answer(False) ↔ EventuallyHasPrimeWitness := by⊢ False ↔ EventuallyHasPrimeWitness
  sorryAll goals completed! 🐙


end Erdos453