Paper - Garner, Lynn - On The Collatz 3n + 1 Algorithm
Paper - Garner, Lynn - On The Collatz 3n + 1 Algorithm
Paper - Garner, Lynn - On The Collatz 3n + 1 Algorithm
Lynn E. garner
Received by the editors April 25, 1980; presented to the Society, January 6, 1980.
1980 Mathematics Subject Classification. Primary 10L10; Secondary 10A40.
© 1981 American Mathematical Society
0002-9939/81/0000-0203/$02.00
19
integers have small stopping times; the above list accounts for 237/256 « 93% of
them. However, stopping times can be arbitrarily large, for o(2" — 1) > 2n. Some
interesting cases of larger stopping times are o(27) = 96, o(703) = 132, o(35,655) =
220, a(270,271) = 311, and ct(1,027,341)= 347. In a computation of stopping times
of integers up to 1,065,000, the largest observed stopping time was 347.
3. A term formula. Let Ck(n) consist of the first k terms of the Collatz sequence
for n. Let m (which depends on n and k) be the number of odd terms in Ck(n), and
let d¡ be the number of consecutive even terms immediately following the ith odd
term. Let d0 be the number of even terms preceding the first odd term. Then the
next term in the Collatz sequence for n is
im m off! —i
**(„) »_£_„+
2k~m
¿ -t^-
,_i2*+1 ■*<•
(1)
We are able to prove that k(/i) = o(n) under certain bound conditions which
arise from a study of the powers of 2 and the powers of 3.
5. Powers of 2 and 3. The behavior of a Collatz sequence is clearly related to the
way in which the powers of 2 are distributed among the powers of 3. We were
surprised to find that the powers of 2 appear to be bounded away from the powers
of 3 by an amount which grows almost as rapidly as the power of 3.
To be specific, we choose M e N, and let
b{M)= max{-log3(l - 2P<J)3-J))
j KM
and
B{M) = max{-log3(2l+i'(')3-> - 1)).
j<M l J
for all m < M. The values of M at which ¿>(M) and B(M) increase are given in
Table 1, with the corresponding values of b and B. (These calculations were carried
out on a Hewlett-Packard HP-19C programmable calculator.)
6. The main theorem. Now we can prove that k(m) = o(n) if the number of odd
terms encountered in the Collatz sequence is not too great. The following theorem
makes the bound condition precise.
by (2). Thus
2 ■*
Now if sk(n) > n, then n < (3m/2k'm)n + (m/3)(l - 3-*"*), and therefore
«<4'«(l-3-é<AÍ>)^r-31'<*í>
3 3
by (3). This leads by the hypothesis to n < n, a contradiction. Hence sk(n) < n,
and er(/i) < k. But k(«) < o(n), so a(/i) = k(/j).
It is conjectured that the bound condition in the theorem is unnecessary, or is
automatically satisfied so that o(ri) = k(/i) in all cases.
7. Application to cycles. Suppose a Collatz sequence enters a cycle which does not
contain 1. Let n be the least term of the cycle; then sk(n) = n for some k E. N.
Hence k(h) < k, so that if o{h) = k(/i), then n is not the least term of the cycle after
all. Thus if stopping time and coefficient stopping time are always the same, then
the only cycle is the 4, 2, 1, . . . cycle.
The same contradiction arises if the number of odd terms in the cycle satisfies
the bound in the theorem. Thus if there is a cycle not containing 1, the number m
of odd terms in the cycle must satisfy m > min{M, (n/2)3,_B(M)(l - 3~b(M))-x},
where n is the least term of the cycle.
Using n > 60,000,000, M = 14,000, B(M) = 8.0, and b(M) = 9.14, we find
m > 13,700, and hence k > 35,400. Thus any cycle not containing 1 must have at
least 35,400 terms.
If the number 2 billion turns out to be the lower bound for n, as alleged, then
M = 41,000,B(M) = 10.2,and b(M) = 9.9 yield m > 40,700and k > 105,000.
In any event, there is only one "short" cycle, the known one.
References
1. C. J. Everett, Iteration of the number-theoretic function fl2n) — n,f(2n + 1) — 3n + 2, Advances in
Math. 25 (1977),42.
2. M. Gardner, Mathematical games, Sei. Amer. 226 (1972), 115.
3. R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976), 241.