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 Mapping,"" 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 It is already known that the 3n+1 Mapping has five cycles. If there were some other cycles, the sixth cycle is not found, yet. I found that a grid chart is useful to detect cycle elements of the an+b Mappings. The result of classifying the cycles, I obtained a rule that the 3n+1 Mapping has not any cycle other than the already known five cycles. So, I got a conclusion that the ""Collatz Conjecture"" is ""True."" Knowing the conclusion, it's too simple, and it's wonder that for over 80 years no one in the world has found such a law. The proof problem of mathematics may be like that. It seems that this is exactly what belongs to the class called ""the Columbus egg"". A manuscript of this paper was submitted to ""The Journal of the Mathematical Society of Japan"" on Feb. 05, 2021.I received an e-mail from the editor on Feb. 12, 2021. ""We regret that we are not able to publish your paper. Such is the pressure on publication space in the journal that we are compelled to return to authors some articles that might have been considered in the past.""It was rejected before it was in the hands of the reviewers.Perhaps submitting to another journal will do the same. If you search the Internet for ""the Collatz problem,"" you can find many pages. However, I couldn't find out if there was a page based on 3 that used the ternary notation system. In light of the content of this paper, it seems that many people were wandering in the wrong direction.I discovered the clues of elucidation earlier than anyone in the world. I understand that this is the only difference. And as soon as possible, I published this paper-back with the feeling that I would like to share this with everyone around the world.By all means, I would like readers to read it carefully as if they were reviewers.

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