Combinatorics and Graph Theory

2109 Submissions

[2] viXra:2109.0071 [pdf] submitted on 2021-09-09 22:01:08

Revised Collatz Graph Explains Predictability

Authors: Steven M. Tylock
Comments: 11 Pages.

The Collatz Conjecture remains an intriguing problem. Expanding on an altered formula first presented in “Bits of Complexity”, this paper explores the concept that “3N + Least-Significant-Bit” of a number allows the complete separation of the step of “dividing by two on an even number” within the Collatz Conjecture. This alternate formula replaces the original Collatz on a one-for-one basis. The breadth and depth of graphs of the resulting path of numbers resolving with this new formula illustrate its fractal nature. Lastly, we explore the predictability of this data, and how the ultimate goal of reaching one prevented previous work from finding the key to understanding 3N+1.
Category: Combinatorics and Graph Theory

[1] viXra:2109.0021 [pdf] replaced on 2022-06-24 10:11:38

On Sums of Product of Powers of Palindromic Sequence and Arithmetic Progression

Authors: Lateef Suaib
Comments: 9 Pages.

In this paper, we combine a real or complex palindromic sequence and arithmetic sequence to produce the sums of product of powers of palindromic-arithmetic sequences. As a result, we generate new expressions for Franel numbers as well as first Strehl identity.
Category: Combinatorics and Graph Theory