Modular Arithmetic Study on Ascending and Descending Operations Related with Collatz Conjecture (2311.12038v1)

Abstract: Sequence of numbers generated by the recurrence relation based on the Collatz conjecture is investigated. An arithmetic operation on the Collatz conjecture is called descending operation, and ascending operation is carried out reversely to the descending operation. Study on the 3x+1 problem against every integer is relevant to that of descending operation against every odd. Any odds can be generated by ascending operations against odds not multiple of 3, and vise versa. Ascending operations against an odd not a multiple of 3 can generate infinite number of odds in the next generation. The odds multiple of 3 are terminal numbers of sequences that no further ascending operations are possible. Descending operations against the odds 1,5,21,.. reach 1, and otherwise reach the odds greater than or equal to 5. No two odds except 1 in each sequence are the same, and the each descending sequence is unique. Every descending sequence can start from terminal numbers, and reaches ultimately the origin of 1 as indicated by the Collatz conjecture. Sequences of odds in the same pattern ignoring the sizes of the odds are generated by ascending operations with the same number of doubling operations against the odds in modular arithmetic.