On Collatz Conjecture

2019/02/15 by Tezcan, Erhan
#11B37 (Primary) #39B05 (Secondary) #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.1902.07312

Abstract

The Collatz Conjecture can be stated as: using the reduced Collatz function C(n) = (3n+1)/2^x where 2^x is the largest power of 2 that divides 3n+1, any odd integer n will eventually reach 1 in j iterations such that C^j(n) = 1. In this paper we use reduced Collatz function and reverse reduced Collatz function. We present odd numbers as sum of fractions, which we call `fractional sum notation' and its generalized form `intermediate fractional sum notation', which we use to present a formula to obtain numbers with greater Collatz sequence lengths. We give a formula to obtain numbers with sequence length 2. We show that if trajectory of n is looping and there is an odd number m such that C^j(m) = 1, n must be in form 3^j×2k + 1, k ∈ ℕ₀ where C^j(n) = n. We use Intermediate fractional sum notation to show a simpler proof that there are no loops with length 2 other than trivial cycle looping twice. We then work with reverse reduced Collatz function, and present a modified version of it which enables us to determine the result in modulo 6. We present a procedure to generate a Collatz graph using reverse reduced Collatz functions.

Related

• 3n+3^k: New Perspective on Collatz Conjecture
• An Exploration Into the Collatz Conjecture with Changed Parameters
• Novel Theorems and Algorithms Relating to the Collatz Conjecture
• Observations towards a proof of the Collatz conjecture using 2^jk+x number series
• About the proof of the Collatz conjecture