A Conjecture on the Collatz-Kakutani Path Length for the Mersenne Primes

TL;DR

This paper proposes a new conjecture linking the path length in the Collatz-Kakutani tree to Mersenne primes, suggesting a proportional relationship with their index, and discusses behavioral differences between Mersenne numbers and primes.

Contribution

It introduces a novel conjecture connecting Mersenne primes with the Collatz-Kakutani problem through a natural path length measure.

Findings

Conjecture that Mersenne prime path length is proportional to its index.

Discussion of behavioral differences between Mersenne numbers and Mersenne primes.

Abstract

We present here a new conjecture for the nature of the Mersenne prime numbers by connecting it with the Collatz-Kakutani problem. By introducing a natural path length on the basis of the Collatz-Kakutani tree, we conjecture that this path length of a Mersenne prime from the root of the Collatz-Kakutani tree is approximately proportional to the index of the Mersenne prime. We also discuss difference of behaviors between Mersenne numbers and Mersenne primes.