I recently came across the Collatz conjecture and have spent some time investigating it. Although I haven’t found a proof of the conjecture I did wonder if a process similar the ‘sieve of Eratosthanes’ (used to isolate the prime numbers) might be used to isolate any integers that didn’t eventually home in on 1. Previous postings have established that we only need be concerned with odd numbers as even numbers divide down to 1 or an odd number so the sieving procedure is as follows:-
Step 1: List all the odd integers from 1 to infinity in column 1.
Step 2: Create a new column, column 2, and record the result of the function 3*n + 1 for each odd integer n in column 1 and reduce the calculated value to its smallest odd number. This obviously takes a long time to do as the column is infinitely long though these are the only calculations that have to be performed! Numbers such as 5, 21, 85 etc. have fallen through the sieve as they go straight to 1 in this step.
Step 3: Create a new column, column 3. Work down column 2 looking at each row in turn. Take the value on row n, column 2 and find the same number in column 1 then record the value found in column 2 at this row, in column 3 of row n. More numbers have now fallen through the sieve leaving fewer odd integers to deal with.
Step 4: Effectively repeat step 3 by forming a fourth column, and look up the values from column 3 in column 1 and record the values found in column 2 in column 3 of the initial row.
I tried to include a table to show how the above process works but, unfortunately, getting it to work in these comments seems to be beyond me at the moment, sorry.
I have tried this process in Excel, with odd integers up to 59995, and the first number which didn’t reach 1 was 703 but this was because it called for a number further down the list than 59995. Just less than 60% of the list went to 1 and this was achieved by column 17 which is much quicker and requires much less computation than the straightforward step by step method.
One thing I think we can conclude is that if there is a smallest number which forms a loop or which never reach 1, then this number must never reach a value in its sequence that is smaller than itself, as this value would, by definition, reduce to 1.
I have also derived formulae which will calculate all the numbers that would eventually lead to any given odd integer. I will post details of this later.