Answer to Question #97811 in Combinatorics | Number Theory for Fayoo

Question

1) in your own words explain collatz conjecture. Have this conjecture been proven?
2) What is the C(n) cycle and the T(n) cycle of the number n= 48?
3) Find the binary encoding of n= 32,53, 80 and explain why they all start with ''111''.
4) What is more common according to the data: r-curves with finite girth or acyclic r-curves?
REFERENCE ARTICLE link: https://arxiv.org/pdf/1811.00384.pdf

Answer 1

1) The Collatz conjecture was not proved yet. It states that, for a positive integer n, an iteration-based sequence built like n/2 when n is even and (3n+1) when n is odd will eventually reach 1.

2)The sequence C(48) is 48 $\to$ 24 → 12 → 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 →1

So, the cycle is 1 → 4 → 2 →1.

 The sequence T(48) is 48 → 24 → 12 → 6 → 3 → 5 → 8 → 4 → 2 → 1.

So, the cycle is 1→2→1.

3)They all start with 111 because when n is greater than or equal with 5, the sequence ends with 8 They all start with 111 because when n is greater than or equal with 5, the sequence ends with 8→4→2→1, which means 3 consecutive applications of T(n) that give even numbers, which means 111 when encoding.

For 32, the sequence is 32→16→8→4→2→1, thus T(32) is always even excepting the last application when it becomes odd (1). Reading from right to left we obtain 011111 which is 11111 as binary encoding sequence.

 

For 53, the sequence is 53→ 80→40→20→10→5→8→4→2→1

The binary encoding sequence, read from right to left, is 0111011110, which is 111011110.

 

For 80, the sequence is 80→40→20→10→5→8→4→2→1.

The binary encoding sequence, read from right to left is 011101111, which is 11101111.

4)the r-curves with no cycles are more common than r-curves with finite girth (40 vs. 3).