The 13x+1 and 11x+ 1 Maps

13x+1

Having looked at the 17x+1 map is my previous post, it was seen at 43 and 61 were special in the sense that they returned to their starting point after 84 steps, whereas every other prime from 17 to 97 terminated in at most 56 steps. Working again in Google Sheets, I applied the 13x+1 algorithm to the primes between 17 and 109. For any integer N>11, the algorithm is as follows:

• IF 2|N THEN N --> N/2 
• IF 3|N THEN N --> N/3 
• IF 5|N THEN N --> N/5 
• IF 7|N THEN N --> N/7 
• IF 11|N THEN N --> N/11 
• ELSE N --> 13*N+1

Here are the results and, for 13x+1, it appears that 19 and 101 are special in that they are the only primes so far that I've found to return to their starting point (151 joins the trajectory of 19 after 10 steps):

• 13 terminates after 25 steps
• 17 terminates after 23 steps
• 19 repeats after 15 steps
• 23 terminates after 6 steps
• 29 terminates after 6 steps
• 31 repeats after 15 steps (overlaps with trajectory of 19)
• 37 terminates after 20 steps
• 41 terminates after 40 steps
• 43 terminates after 7 steps
• 47 terminates after 28 steps
• 53 terminates after 10 steps
• 59 terminates after 10 steps
• 61 terminates after 10 steps
• 67 terminates after 60 steps
• 71 terminates after 6 steps
• 73 repeats after 15 steps (overlaps with trajectories of 19 and 31)
• 79 terminates after 43 steps
• 83 terminates after 8 steps
• 89 terminates after 37 steps
• 97 terminates after 24 steps
• 101 repeats after 15 steps (overlaps with trajectories of 19, 31 and 73)
• 103 terminates after 64 steps
• 107 terminates after 12 steps
• 109 terminates after 56 steps

The 19, 31, 73 and 101 sequences overlap as the diagram below shows (progression is anticlockwise):

Further investigation has shown that 151 joins the trajectory of 19 after 10 steps. Primes 113, 127, 131, 137, 139 and 149 all terminate. It will be interesting to investigate whether other non-terminating primes join this above circle.

My conjecture is that all numbers, under the 13x+1 algorithm, will either enter this loop or else get reduced to 1.

11x+1

For the 11x+1 map, the algorithm to be applied is that for any integer N>7:

• IF 2|N THEN N --> N/2 
• IF 3|N THEN N --> N/3 
• IF 5|N THEN N --> N/5 
• IF 7|N THEN N --> N/7  
• ELSE N --> 11*N+1

Once again, most numbers terminate but there is a cycle as shown below, starting at 17 and proceeding in an anticlockwise direction:

My conjecture is that all numbers, under the 11x+1 algorithm, will either enter this loop or else get reduced to 1.

What I am seeing thus far with the Px+1 map is that values of P=17, 13 and 11 will reduce most numbers to 1 but in each case there is a set of prime numbers that form part of a cycle. I haven't tested for P=7 or 5 yet. However, we know that for P=3 there is no known cycle of this nature.