znam_problem

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Znám's problem: For each k ≥ 2, find all sets of k distinct integers greater than 1 such that each element divides the product of the others plus 1. Solutions known for small k; the complete classification for all k is open.

Why this node was retired

The posted statement is

import Mathlib

theorem znam_problem (k : ℕ) (hk : 2 ≤ k) :
    ∃ (x : Fin k → ℕ),
      (∀ i j : Fin k, i ≠ j → x i ≠ x j) ∧
      (∀ i : Fin k, ∀ j : Fin k, i ≠ j → x i ∣ (∏ l : Fin k, x l) + 1) := by
  sorry

Each divisor is tested against the product of all entries, rather than the product of the other entries.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require x_i>1 and x_i to be a proper divisor of1+∏_(j≠i)x_j for the standard Znám problem. Ask classification/existence only for its admissible sizes, rather than asserting the unsupported all-k≥2 existence claim.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.