Abstract:

The aim of our project is to investigate the \(3n + 1\) conjecture. It is very hard to give a general path for each natural number to arrive at \(1\). So we investigate its negation i.e. there exists a natural number \(k\) with no path to \(1\). There are two possibilities: either \(k\) takes a path which becomes a cycle to after \(n\) steps, or its path is increasing indefinitely. These two possibilities lead us to study pre-numbers of any odd natural number and the number of peaks of paths. In the project, several interesting results were obtained by studying backward paths, number of peaks and cycles or forward paths.

Firstly, we tried our best to trace back the path by studying all the pre-numbers of any natural number. We successfully showed that every odd number not divisible by \(3\) has infinitely many pre-numbers by finding them explicitly, and hence obtained a beautiful known result: the odd numbers \(A\) and \(4A + 1\) fall into the same number. Moreover, we also found that two third of these pre-numbers has infinitely many pre-numbers. Continuing in this way, we constructed a decreasing path of any length to arrive at any given number. This leads to a beautiful corollary: there are infinitely many distinct decreasing paths of different lengths to \(1\).

Secondly, we studied the peaks of the path of any number and obtained a theorem that for any T-number \(k\) and natural number \(r\), there exists a path to \(k\) of length \(2r\) with exactly \(r\) peaks.

Thirdly, we assumed, on the contrary, that there is a path with the same beginning and ending, and obtained a constraint on both the length and the sum of powers of \(2\).

Fourthly, we found infinitely many pairs which meet before \(1\) and fall into the same path afterward.

Finally, after investigating the possibilities of general \((a, b, c)\)-Conjectures, we concluded that the only possible conjectures are \(3n + b\) Conjectures.

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