Ordowski’s Immediate-Successor Weak Pseudoprime Conjecture
Abstract
Ordowski’s immediate-successor weak-pseudoprime condition holds exactly at odd composite successors.
Theorem 1.1 (Odd composite successors are exactly the qualifying successors).
Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/OrdowskiImmediateSuccessorWeakPseudoprime.result (✓ std3). ∎
Resolves. Problems/oeis-a239293-ordowski-immediate-successor-weak-pseudoprime (proved) by D5/S3/Arith/Congruence/OrdowskiImmediateSuccessorWeakPseudoprime.result.
Source. Repository-derived.
Acknowledgement. Robert Ferreol; Thomas Ordowski (2018). OEIS A239293, Smallest composite c > n such that n^c == n (mod c). URL: https://oeis.org/A239293.
Commentary.
For every natural n at least one, the composite successor n+1 satisfies the weak-pseudoprime congruence exactly when n+1 is odd. Since n+1 is the immediate successor of n, qualification at this modulus is already least among qualifying composites greater than n. Modulo n+1, the residue of n is minus one, so its power is classified by the parity of n+1. An even qualifying successor would make minus one equal to one; compositeness excludes the only boundary modulus two. The underlying casts, parity powers, and residue facts are pinned Mathlib material.