The PrimeLatz Conjecture

The PrimeLatz Conjecture is so named in deference to and because of its similarity to the Collatz Conjecture that I've written about in earlier posts, namely:

• The 3x+1 Problem on the 1st November 2016
• The Collatz Conjecture Revisited on the 15th March 2018
• The 17x+1 Map on the 18th March 2018

The so-called Collatz Trajectory refers to the sequence of numbers generated by the following rule. Start with any positive integer. If the number is even, divide it by 2. If the number is odd, multiply by 3 and add 1. The conjecture is that the sequence of numbers thus generates will always lead to 1. So far, no exceptions have been found.

The PrimeLatz Trajectory is similar except that for odd numbers, the rule is to add the next three primes to the number. This will always generate an even number that is then divided by 2. The PrimeLatz Conjecture is that the sequence of numbers thus generated will always lead to a loop.

Take the case of 82. Here is the sequence:

82 41 184 92 46 23 120 60 30 15 74 37 168 84 42 21 104 52 26 13 72 36 18 9 50 25 122 61 272 136 68 34 17 88 44 22 11 60 ...

The sequence repeats after 11 is reached. Because 11 is odd, the next three primes are added. Thus 11 + 13 + 17 + 19 = 60 but 60 was reached earlier when 120 was divided by 2. Let's take another example, this time 84. The sequence is:

84 42 21 104 52 26 13 72 36 18 9 50 25 122 61 272 136 68 34 17 88 44 22 11 60 30 15 74 37 168 84 ...

Most numbers fall into a loop fairly quickly but what's of interest are those numbers that are more stubborn, such as 83. I deliberately chose numbers that were one above and one below 83 to emphasise the difference. Here is what the OEIS says about the behaviour of 83 where \(a(n)\) represents the sequence of terms:

Most small initial values have a very small orbit of few more than the 30 elements of the loop. 

N = 83 = a(0) is the most remarkable exception (having an orbit of 16180 + 30 elements), which motivates this sequence.  

N = 443 = A293978(0) is another exception, with an orbit of 9066+30 elements, and N = 209 also has a comparatively large orbit of 941 + 30 elements, distinct from those of 83 and 443.

The initial value a(0) = 83 is odd, so we add to 83 the next 3 primes (89, 97 and 101) to get a(1) = 370. 

370 is even, so we divide by 2 to get a(2) = 185, and so on. 

After 8337 iterations, we get a(8337) = 10780054699424618132644155893087038044817868609971935265882538442720. 

This is the largest value we will reach. Since this is even we divide by 2 to get a(8338). The result a(8338) is again even, so we divide by 2 once more to get a(8339), and so on... 

After iteration 16171, we reach a(16171) = 768. The next 8 iterations consist in dividing by 2, until we get a(16179) = 3. Since this is odd, we add the next three primes (5, 7 and 11) to reach a(16180) = 26 = A193230(14). This is an element of the loop: 30 iterations later, we get again 26, and the sequence has become periodic.

All numbers less than 100 million eventually fall into the following loop:

9,50,25,122,61,272,136,68,34,17,88,44,22,11,60,
30,15,74,37,168,84,42,21,104,52,26,13,72,36,18

ADDENDUM 10th May 2020:

I'm now approaching my 26000th day on Earth and I've noticed that, in the orbit of 83, there is a cluster of numbers between 25000 and 31000 with wide gaps on either side. Figure 1 shows what I mean:

Figure 1

There is a gap of over 10000 from 15456 to 25710 and then a gap of over 20000 from 30912 to 51536. As I write this on the 10th May 2020, I'm 25970 days old and this number is in the 83 orbit. There are a total of 54 numbers that fall in the range from 25710 to 30912.

The full orbit of 83 can be viewed, in comma-delimited format, by following this link (it's 176 pages long):

https://docs.google.com/document/d/1uP-sBqDNjbhkEqK8IEDG-doN2QeYXnq0K8QObWkzPWU/edit?usp=sharing