Download Digit Characteristics in the Collatz 3n+1 Iterations

Digit Characteristics in the Collatz 3n+1 Iterations
Subhash Kak
September 24, 2014
Abstract: The trajectory of the 3n+1 problem is generated when an odd number n is replaced
by 3n+1 and even number is halved in an iterative manner leading ultimately to 1. Some new
properties of these sequences are presented. In particular, it is shown that numbers that are
equal to 1 modulo 4 iteratively converge to 1 although we do not specify the number of steps
needed. It is shown that for the numbers tested, the proportion of prime numbers in the
sequences is quite high.
Introduction
The 3n+1 problem is associated with many names that generally begin with L. Collatz (1937)
and includes other mathematicians [1]-[4]. According to this problem, take any natural
number n, if n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to
obtain 3n + 1 and repeat the process indefinitely. The conjecture is that no matter what
number one starts with, one will eventually reach 1.
Formally, the mapping is the function s: N → N on the set of positive integers:
 n/2
s ( n)  
3n  1
if n is even
(1)
if n is odd
Let s 0 (n)  n and s k (n)  s(s k 1 (n)) for k  N . This will generate the orbit of n.
If n = 7, one gets the sequence 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1; in
other words, the orbit or trajectory of 7 has 17 elements and the path length is 16. The
maximum value generated is 52. The trajectory of n = 3 is 3, 10, 5, 16, 8, 4, 2, 1, that is it has
8 elements in it or it has a path length of 7, with a maximum value of 16. The Collatz
sequence for n = 25, takes 23 steps, climbing to 88 before descending to 1: ( 25, 76, 38, 19,
58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 ). The algorithm for the
sequence can also be written so that it excludes even numbers, but we shall not do so in this
paper.
The questions that interest the computer scientist are: the characteristics of the numbers in the
orbit, the path length, the largest number encountered in the path, and the proportion of
numbers that is prime. From the perspective of cryptography, one is also interested in the
randomness characteristics of the numbers in the path either directly or when the sequence is
reduced to a binary sequence by replacing the even numbers by -1 and odd numbers by +1
(that is equivalent to considering the numbers mod 2).
Our expectation is that the randomness characteristics [5],[6] of the sequence will be
excellent due to the complexity of the iteration graph. We like to point out the related
question of randomness in physical systems [7]-[9] that may also be attributed to complexity.
By way of contrast we can look at d-sequences where for the expansion 1/p for prime p, the
binary sequence a(n) is given by the formula [10]-[11] a(n)  2n mod p mod 2 . In contrast to
the successive multiplication by 2 in the d-sequence, we have successive multiplication by 3
(together with addition of number 1 and reduction of even numbers) for the Collatz sequence.
1
The d-sequence thus does not appear to have the same “information content” as the Collatz
sequence. Two other number theoretic random sequences are related to the generation of
Pythagorean triples [12] and the divisor pairs function [13]. The difficulty of establishing the
Collatz conjecture is due to the “random” nature of its graph [14] (Figure 1)
The Collatz map can also be viewed as a dynamical system and various generalizations of
this map have been analyzed [15].
17
53
96
1024
52
160
48
512
26
80
24
256
13
40
12
128
20
6
64
10
3
32
5
9
336
28
168
85
84
42
21
16
8
4
2
1
Figure 1. A portion of Collatz mapping as a directed graph
In this paper we summarize some general properties of the Collatz sequence. In particular, we
present results on odd counts and iterative convergence.
Path length, odd count, and primes
We first summarize in Tables 1 and 2 characteristics of Collatz sequences with respect to the
maximum value and the longest path for n < 1000.
Table 1. Sequence of maximum values up to n < 1,000
n Path length
Odd count
Max value
1
0
0
1
2
1
1
2
3
7
3
16
7
16
6
52
15
17
6
160
27
111
42
9,232
255
47
16
13,120
447
97
35
39,364
639
131
48
41,524
703
170
63
250,504
2
As we can see, the path length does not grow quickly as n increases and the maximum value
increases much faster than the path length.
Table 2. Sequence of longest paths up to n < 1,000
n Path length
Odd count
Max value
1
0
0
1
2
1
1
2
3
7
3
16
6
8
3
16
7
16
6
52
9
19
7
52
18
20
7
52
25
23
8
88
27
111
42
9,232
54
112
42
9,232
73
115
43
9,232
97
118
44
9,232
129
121
45
9,232
171
124
46
9,232
231
127
47
9,232
313
130
48
9,232
327
143
53
9,232
649
144
53
9,232
703
170
63
250,504
871
178
66
190,996
In Table 2 the sequences are listed according to the largest path length. It is striking that the
maximum value for many of these sequences is the same: two sequences have value 16, three
have 52, and ten have value 9,232. It is clear that sequences with the same maximum number
are related.
We now provide results on the number of primes with respect to n in Table 3.
Table 3. Sequence of most primes up to n < 10,000
n Path length
Odd count
Prime count
1
0
0
0
2
1
1
1
3
7
3
3
7
16
6
6
19
19
7
7
27
111
42
25
97
118
44
26
231
127
47
29
487
142
52
30
1153
151
55
31
1407
172
63
32
1,895
175
64
35
2,539
178
65
36
4,011
189
69
37
6,171
262
97
43
3
A large proportion (56%) of the odd numbers in the range 1 ≤ n ≤ 10,000 is prime. This
suggests that the Collatz sequence can be used to generate numbers that have a high
probability of being prime.
General properties of Collatz sequences
First note that one need only consider odd numbers in the transformation as the even numbers
are divided successively by 2 until reduced to an odd number. If the number is n mod 4  1,
then n = 4k+1. Upon use of (1), the next number is n  3(4k  1)  1  12k  4. When reduced
further, it becomes n  3k  1 .
If k is odd, n is even and upon further reduction it may lead to numbers that are congruent to
either 1 or 3 modulo 4. If k is even, the n values are alternately 3 and 1 modulo 4. This
means that if one can show that n mod 4  1 numbers converge, so will the n mod 4  3
numbers.
Lemma. If an odd number a leads to another odd number (after several applications of the
Collatz transformation) b, then 4a+1 also leads to b.
Proof. The odd number a will go to 3a+1, whereas the number 4a+1will go to 12a + 4 which
when reduced will also equal 3a+1.
As example, 3 goes through 10 to 5; so will 13, 53, 213, 853, 3,413, 13,653, .. go to 5 after
passing through intermediate even numbers.
If one were ready to skip intermediate odd numbers, then one can consider the general linear
transformation on a that takes it to ca+d. So long as this number is odd, it will be
transformed further by (1) that we will put equal to some product of the original transformed
number 3a+1. This means:
3ca  3d  1   (3a  1)
(2)
Here α is a suitable constant. This, in turn, gives the conditions:
3ca  3a and 3d  1  
(3)
In other words, c    3d  1. By taking different values of d=1, 2, 3, 4, 5, … we can
generate maps between different starting numbers that lead to the same terminal odd number.
The transformations, thus, will be:
4a+1
7a+2
10a+3
13a+4
4
…. and so on.
If a takes the value 1, then the numbers 5, 9, 13, 17, … will belong to the same class. This is
the class of numbers n mod 4  1, and all these numbers will eventually reach 1 since we
know elements of this class for which this is true. This argument does not tell us what the
length of the path for any specific number will be. Other classes are 13, 23, 33, 43, … when
a=3, and 21, 37, 53, 69, … for a=5, and so on.
The only other odd numbers that one needs to account for are those for which n mod 4  3.
But we have already seen that in their evolution they alternate with n mod 4  1 numbers.
Probabilistically, there is probability ½ that the next number will be of the type n mod 4  1.
In other words, at the end of a string of k-1 odd numbers of type n mod 4  3 there is the
probability 2  k that the next number is of type n mod 4  1 and the string will converge.
Summarizing,
Theorem 1. All numbers n mod 4  1 iterate to 1 although the path length cannot be
specified. Numbers n mod 4  3 iterate to 1 with probability equal to 1  2  k where k is the
count of such numbers in the string.
Proportion of odd numbers
For a large set of random odd numbers k, k/2 are divisible of 2 and not by powers of 2, k/4
are divisible by 4 and not by powers of 4, k/8 are divisible by 8 and not by powers of 8, and
so on. The numbers that are divisible by 2 and not by its powers return to an odd number
right away by the halving process of (1). Those that are divisible by 4 and not by powers of 4
return in 2 steps, and those that are divisible by 8 and not its powers return in 3 steps, and so
on. Therefore, for k odd numbers, the number of corresponding even numbers is:
k 2k 3k 4k 5k
(4)
S 



 ...
2 4
8 16 32
Multiplying (4) by ½ and subtracting from itself, we get:
S k k k k
     ...
2 2 4 8 16
(5)
It follows that the count of even numbers is 2k.
Theorem 2. The count of odd numbers in a large path Collatz sequence is approximately
one-half the count of even numbers.
Conclusion
This paper has presented several properties of Collatz sequences. In particular, it is shown
that numbers that are equal to 1 modulo 4 iteratively converge to 1 although we do not
specify the number of steps needed. Numbers that are equal to 3 modulo 4 iterate to 1 with
probability equal to 1  2  k where k is the count of such numbers in the string being
5
considered. It is shown that for the numbers tested, the proportion of prime numbers in the
sequences is quite high.
References
1. L.E. Garner, On the Collatz 3n + 1 algorithm. Proceedings of the American
Mathematical Society 82 (1): 19–22 (1981)
2. B. Hayes, On the ups and downs of hailstone numbers. Scientific American, January
1984.
3. S. Andrei and C. Masalagiu, About the Collatz conjecture. Acta Informatica 35: 167179 (1998)
4. J.P. Van Bendegem, The Collatz conjecture: A case study in mathematical problem
solving. Logic and Logical Philosophy 14: 7–23 (2005)
5. A Kolmogorov, Three approaches to the quantitative definition of information.
Problems of Information Transmission. 1:1-17 (1965)
6. S. Kak, Classification of random binary sequences using Walsh-Fourier analysis.
IEEE Trans. on Electromagnetic Compatibility, EMC-13: 74-77 (1971)
7. S Kak, Information, physics and computation. Found. of Phys. 26: 127-137 (1996)
8. R. Landauer, The physical nature of information. Physics Letters A 217: 188-193
(1996)
9. S Kak, Quantum information and entropy. Int. Journal of Theo. Phys. 46: 860-876
(2007)
10. S. Kak and A. Chatterjee, On decimal sequences. IEEE Trans. on Information Theory
IT-27: 647 – 652 (1981)
11. S. Kak, Encryption and error-correction coding using D sequences. IEEE Trans. on
Computers C-34: 803-809 (1985)
12. S. Kak and M. Prabhu, Cryptography applications of primitive Pythagorean triples.
Cryptologia 38: 215-222 (2014)
13. S. Kak, Random sequences based on the divisor pairs function. arXiv:1210.4614
(2012)
14. J.C. Lagarias, The 3x + 1 problem and its generalization. Amer. Math. Monthly,
92(1): 3-23 (1985)
15. G.J. Wirsching, The Dynamical System Generated by the 3n+1 Function. Lecture
Notes in Mathematics 1681. Springer-Verlag (1998)
6