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Sequence of numbers generated by the recurrence relation based on the Collatzconjecture is investigated. An arithmetic operation on the Collatz conjectureis called descending operation, and ascending operation is carried outreversely to the descending operation. Study on the 3x+1 problem against everyinteger is relevant to that of descending operation against every odd. Any oddscan be generated by ascending operations against odds not multiple of 3, andvise versa. Ascending operations against an odd not a multiple of 3 cangenerate infinite number of odds in the next generation. The odds multiple of 3are terminal numbers of sequences that no further ascending operations arepossible. Descending operations against the odds 1,5,21,.. reach 1, andotherwise reach the odds greater than or equal to 5. No two odds except 1 ineach sequence are the same, and the each descending sequence is unique. Everydescending sequence can start from terminal numbers, and reaches ultimately theorigin of 1 as indicated by the Collatz conjecture. Sequences of odds in thesame pattern ignoring the sizes of the odds are generated by ascendingoperations with the same number of doubling operations against the odds inmodular arithmetic.

Modular Arithmetic Study on Ascending and Descending Operations Related with Collatz Conjecture