On the nonexistence of cycles for the Collatz function

TL;DR

This paper proves that the Collatz function has no cycles other than the well-known (4, 2, 1) cycle, supporting the conjecture that all sequences eventually reach this cycle.

Contribution

It establishes a proof that no other cycles exist for the Collatz function, advancing understanding of the conjecture's validity.

Findings

No additional cycles for the Collatz function exist

Supports the conjecture that all sequences reach (4, 2, 1)

Provides a theoretical proof for cycle nonexistence

Abstract

The Collatz function is defined as C(n) = n / 2 if n is even and C(n) = 3n + 1 if n is odd. The Collatz conjecture states that every sequence generated by the Collatz function ends with the cycle (4, 2, 1) after a finite number of iterations. In this paper it is shown that there exists no other cycle for the Collatz function.