Overview

The Collatz Conjecture is restricted to positive area of 3n+1 Mapping. Since I applied it to include whole integers (negative and positive integers and zero) of 3n+1 Mapping and also to 3n+b Mappings, I determined many rules, such as Odd Number's Relation Rule, Rule to Form Tree Structures and Rule to Form Cycles. Also I expanded the Collatz Conjecture to an+b Mappings which Thwaites indicated, then I obtained generalized rules. If the problem of the Collatz Conjecture was 10n+6, ordinarily the decimal number system would be used. If the number was odd, the decimal point would be moved to the right side for one digit, and 6 would be placed to fill the space at the unit digit position. I used the radix a notation system for the an+b Mapping. If the value of b is indicated by a single digit, the value of b is used to fill the space at the unit digit position. If it is an even number, half the number will be placed one step down. When the halved number becomes an odd number, it will be replaced with an even number as described above. Therefore, even numbers in the radix a notation are lined up in each row. As a result, even numbers are arranged like a belt, making them look like a continent or a long island on a map. The west coast line is made by an arrangement of the most significant non-zero digits of integers. Then, I found out that the west coast line seems like one straight inclined line. I obtained a theory that if a>4, then an+b Mappings are divergent, and if a

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