TITLE:
Base-X Conjecture and Collatz Conjecture Proof
AUTHORS:
Sheng Zhao
KEYWORDS:
Collatz Conjecture, Base-X Conjecture, Collatz Spiral, Collatz Ring, Collatz Tree
JOURNAL NAME:
Applied Mathematics,
Vol.16 No.4,
April
28,
2025
ABSTRACT: A new Base-X Conjecture was introduced in this paper, and Collatz Conjecture is just one case of Base-X Conjecture - Base-3 (Ternary). Based on Base-X number system property and Collatz Conjecture iteration, it has been proved that for any positive integer D, there are n and m which exist for
D
n
+
Y
n
=
2
m
.
D
n
+
Y
n
is just the result built up by collecting divided by 2 of Collatz Conjecture iteration. Divided by 2m will make the Collatz Conjecture get a result of 1 for any positive integer. Also, the Collatz Tree showed that for any odd positive number, there is only one route existing in the Collatz Tree down to 1 on Collatz Conjecture iteration.