Nonexistence of Nontrivial Cycles in the Collatz Conjecture: A Modular and 2-Adic Approach

Rigorous proof that the Collatz function T(n)=(3n+1)/2v2(3n+1)T(n)=(3n+1)/2v2(3n+1) admits no nontrivial cycles among positive integers. The method combines recursive identities, modular analysis, and 2-adic properties to show that any hypothetical cycle leads to contradictions in congruences modulo powers of 2. Mathematical codes: 11B37 • 11A07 • 11B50 • 11D88 • 05C20 Keywords: Collatz conjecture, cycles, 2-adic valuation, modular analysis, modular graphs, exponential Diophantine equations.

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