partition_congruences_conjecture

⚠️ 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.

Partition congruences: Beyond Ramanujan's p(5n+4)≡0(mod 5), p(7n+5)≡0(mod 7), p(11n+6)≡0(mod 11), do other systematic congruences exist for all primes? Ono proved infinitely many congruences mod any prime ≥ 5. Complete characterization open.

Why this node was retired

The posted statement is

import Mathlib

theorem partition_congruences_conjecture :
    ∀ p : ℕ, Nat.Prime p → 5 ≤ p →
    ∃ (a b : ℕ), 1 ≤ a ∧ 0 ≤ b ∧ b < p ∧
    ∀ n : ℕ,
      (∑ k ∈ Finset.range (n + 1),
        (Nat.choose (n + k) k : ZMod p)) = 0 := by
  sorry

The binomial sum is not the partition function and does not contain either chosen progression parameter.

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

Proposed corrected statement

Define the partition-number function P and ask for appropriate progression parameters A,B with P(A*n+B)≡0 modp for all n, using the exact progression/prime-power conditions of the intended source theorem.

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.