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Erdős problem 1056

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Let k≥2k ≥ 2. Does there exist a prime pp and consecutive intervals I0,…,IkI_0,\dots,I_k such that ∏n∈Iin≡1mod  n\prod\limits_{n{\in}I_i}n \equiv 1 \mod n for all 1≤i≤k1 \le i \le k?

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4 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

1056.lean

Retained formal statement4 of 4

Noll and Simmons asked, more generally, whether there are solutions to q1!≡⋯≡qk!mod  pq_1! \equiv \dots \equiv q_k! \mod p for arbitrarily large kk (with q1<⋯<qkq_1 < \dots < q_k).

FormalConjectures/ErdosProblems/1056.leanErdos1056.erdos_1056.variants.noll_simmons6 linesExact file
True ↔  ∀ᶠ (k : ℕ) in Filter.atTop,    ∃ p,      ∃ (_ : Nat.Prime p),        ∃ Q,          ∃ (_ : StrictMono Q) (_ : ∀ (i : Fin k), Q i < p), ∀ (i j : Fin k), (Q i).factorial ≡ (Q j).factorial [MOD p]
OpenStatement only, no proof