On Collatz conjecture

AbstractThe two calculations of Collatz conjecture are as follows.(1). If a number is odd, triple it and add one.(2). If a number is even, divide it by two.If a number is odd, the calculation (1) above is performed first, then the calculation (2) above is executed consecutively afterthe calculation (1) as the result number of the calculation (1) is always even.In this paper, the successive calculations of (1) and (2) are treated as a single set of calculations and this one set of calculations,referred to as“ the set of calculations”.When an odd number before the set of calculations is denoted as 2n + 1, the numberafter the set of calculations is 3n + 2.So the number after the set of calculations is greater than the odd number before the set of calculations.If the number after this set of calculations is odd again, you can perform this set of calculations once more, and the numberafter the next set of calculations will become even larger. Starting with an original odd number, you can repeat the set ofcalculations as long as the result is an odd number, causing the original odd number to increase progressively.In this proof paper, we define a set of calculations as successful if the resulting number from the set of calculations is odd.We consider how many times the set of calculations can be repeated; in other words, how large the original odd number canbecome.We perform the first set calculations on original odd number sequence. Then, we exclude even numbers from the second-levelsequence, which consists of the numbers resulting from the set of calculations, and create the second-level odd number sequence.This process is repeated as long as odd numbers remain after the set of calculations.When we take the smallest term that remains after the each set calculations and create a sequence using the original oddnumbers corresponding to the smallest term that remains after each set calculations, we obtain the following sequence.Recurrence relation:Initial term a1 = 3Recursive formula: aK+1 = 2ak +1where k represents the number of successful repetitions of the set of calculationsThe general term of this sequence is expressed as follows:an = 2n -1(n = k+1, where k represents the number of successful repetitions of the set of calculations)The sequence above is a sequence of Mercenne numbers.Because this sequence diverges to infinity, Collatz conjecture does not hold.(You can repeat the set calculations to infinity.)