Collatz Conjecture: 3n+5

In my previous articles on the Collatz Conjecture, I've looked at 3n+1 (the well-known original); then 5n+1 and 3n+3, referring to what to do when a term in the series is odd.  In this article, I'll look at 3n+5

Let's start with the smaller numbers and see where they take us:

Collatz:  3n+5

Loop A:  lowest = 1

9, 32, 16, 8, 4, 2, 1,8,4,2, 1, 8...

Loop B: lowest = 3, loop goes around a minimum of 19. A frequent entry path is via 7, 11 or 13.
24, 12, 6, 3,14,7,26,13,44,22,11,38,19,62,31,98,49,152,76,38,19...

Loop C:  lowest = 5.  The paths to this loop contain the multiples of 10.
15, 50, 25, 80, 40, 20, 10, 5,20,10,5, 20...

Loop D:  lowest = 23.  Also contains 29.
23,74,37,116,58,29,92,46,23, 74 ...

Other paths into the loops:

17, 56, 28, 14 --> loop B

27, 86, 43, 134, 67, 206, 103, 314, 157, 476, 238, 119, 362, 181, 548, 274, 137, 416, 208, 104, 52, 26, 13, --> Loop B:

29 = loop D

31 = loop B

33, 104, 52, 26, 13 --> Loop B

35, 110, 55, 170, 85, 260, 130, 65, 200, 100, 50, 25, 80 --> Loop C

37, 116, 58, 29 --> Loop D

39, 122, 61, 188, 94, 47, 146, 73, 224, 112, 56, 28, 14, 7 --> Loop B eventually

Summary

As with anything Collatz-related, it's easy to show results but difficult to prove anything.  One thing I can say is that I've only identified four loops with most of the numbers up to 100, and it's highly likely that this will hold for larger numbers, although they may take longer paths.

My articles on the Collatz Conjecture:

Snakes and Ladders (an introduction to the Collatz Conjecture)
Collatz: 5n + 1
Collatz:  3n + 3
Collatz: 3n + 5
Generative AI proves the Collatz Conjecture (published 1 April 2024)