🧵 5x+1

Anonymous at Tue, 30 Apr 2024 00:22:46 UTC No. 16152486

Recently, I've heard of the Collatz Conjecture (3x+1), and was intrigued by it. I then decided to test out the (5x+1) pattern, and found these results:

For the (5x+1) pattern, take a number, for example 5. If the number is odd, do 5x+1. If it's even, do x/2. Then keep on doing it. You'll notice that the numbers eventually fall into a loop. For 3x+1, the loop was 4,2,1. For (5x+1), however, for numbers up to 32011, numbers can fall into SEVEN unique loops. The names of the loops will be denoted by the smallest number of each loop. These loops are {1.0, 7.0, 13.0, 17.0, 121.0, 1151.0, 2037.0}. Now here's the question:
1. Are these the only loops possible?
2. Is there a recognizable numerical relationship between the starting number and the loop it falls into?
3. Can we predict the number of unique loops possible for patterns kx+1, with k being an odd number greater than 1?