What question did this study set out to answer?

This research aims to provide a complete proof of the Collatz conjecture through novel methods of classification and measurement.

June 13, 2026Open Access

The Collatz Conjecture: A Descent Proof via Residue Classes and a New Binary Measure

Key Points

This research aims to provide a complete proof of the Collatz conjecture through novel methods of classification and measurement.
Classified odd numbers based on their residue classes modulo 8.
Developed a new decreasing measure for class 7 mod 8 using the number of trailing ones in binary representation.
Applied modular arithmetic and the well-ordering principle to establish the proof.
The measure τ (k) strictly decreases while numbers remain in the 7 (mod 8) class.
Upon leaving the 7 (mod 8) class, numbers quickly transition to 1 (mod 8) or 5 (mod 8), both of which are strictly decreasing.
All paths are shown to eventually lead to 1, confirming the conjecture's validity.

Abstract

The Collatz conjecture (3x + 1 problem) has remained unproven for over 85 years.This paper presents a complete proof based on an elementary classification of oddnumbers by their residue modulo 8 (1, 3, 5, 7) and a new decreasing measure for themost elusive class 7 (mod 8): the number of trailing ones in the binary expansion ofk, where n = 8k + 7. While a number stays in the 7 (mod 8) class, τ (k) strictlydecreases, forcing the chain to be finite. Once it leaves, it quickly reaches a 1 (mod 8)or 5 (mod 8) number, both of which are strictly decreasing and therefore must reach1 by the well-ordering principle. The proof is elementary, self-contained, and usesonly modular arithmetic, binary representation, and the well-ordering of the positiveintegers.

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