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

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

2011-04-13

arxiv.org/abs/1104.2804v2

Mathematics

Number Theory

9 pages, 4 figures, submitted for a publication

Scientific paper

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 \textit{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.

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