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We present a complete and elementary proof of the Collatz conjecture. The approach relies on reducing the problem to odd numbers via the Syracuse function and classifying all odd numbers greater than one into a finite set of residue classes modulo powers of two. For each of these classes, we explicitly demonstrate that a strictly decreasing inequality S k (n) < n holds for some k ≤ 6. This guaranteed descent, combined with induction, implies that every trajectory eventually reaches the trivial cycle {1, 4, 2}. The proof uses only basic modular arithmetic and is fully verifiable by hand
stavitsky A. D. 2026. An Elementary Proof of the Collatz Conjecture via Finite Residue Class Covering. PREPRINTS.RU. https://doi.org/10.24108/preprints-3114927
