Number theory
The first 20 terms of a series with 𝑎 0 =7
𝑛
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19
𝑎 𝑛
7 22 11 34 17 52 26 13 40 20 10 5
16 8
4
2
1
4
2
1
The conjecture ultimately suggests that for any starting term 𝑎 0 , the series converges at 1.
Advancements in technology have allowed supercomputers to compute the first 2.95 x 10 20 series, yet no proof has been found to generalize the result. Indeed, the conjecture followed simple rules and yet displayed chaotic behaviour. If we had set 𝑎 0 =27 , the series would have climbed up to 9000 before returning to 1. The unpredictability of the function made it extremely resistant to modern mathematical tools such as induction. The sequence is not monotonic: some numbers grow dramatically before eventually shrinking. Another approach would be probabilistic modelling. If one assumes that each ‘odd step’ roughly multiplies the current value by 1.5 (after multiplying by 3 and adding 1 the number will become even), while each ‘even step’ divides by 2, then on average the sequence appears to drift downward, suggesting a convergence to 1. However, this reasoning is more heuristic than rigorous: individual trajectories can rise to enormous values before falling, and such fluctuations cannot be ruled out with probability arguments alone. The problem was so strenuous that professionals in the fields of number theory warned any self-respecting mathematicians to stay away from the problem. Hungarian mathematician Paul Erdős famously said: ‘Mathematics is not yet ready for such problems’, 2 suggesting that it required some new, unknown mathematical tool to be proven. More recently, I encountered an informal argument posted online, with an anonymous user claiming that they had found an unofficial proof for the conjecture. The post visualizes the conjecture as a ‘Collatz Tree’. 3 Being told that a proof for the problem would be too advanced for our current understanding, the post piqued my interest and led me to investigate further. The proof (unofficial) Essentially, all positive integers can be expressed in the form 2 𝑛 𝑎 , with a being an odd positive integer. Therefore, all positive integers can be mapped onto different branches, with each branch starting from a, then 2a, 4a, etc. Then, we further map all odd numbers 𝑎 to their counterparts 3𝑎+1 .
2 https://www.youtube.com/watch?v=094y1Z2wpJg . 3 https://www.reddit.com/r/Collatz/comments/1b2vrhi/the_unofficial_proof_of_the_collatz_conjecture .
66
Made with FlippingBook - PDF hosting