The Collatz Problem
Ellen Dickerson
Introduction to Collatz Conjecture
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Named after Luther Collatz
Proposed in 1937
Unsolved for over 60 years
Also known as a bunch of other things
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Wondrous numbers
3n+1 conjecture
Ulam conjecture
And many more
What is the Collatz Conjecture?
Take any natural number n. If n is even, divide n by
2 to obtain (n/2). If n is odd, multiply it by 3 and
add 1 to obtain 3n+1. Repeat the process
indefinitely. The Conjecture is that no matter what
n you start with, you will always eventually reach 1.
ex: try doing this starting with n=3
n=27
• { 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214,
107, 322, 161, 484, 242, 121, 364, 182, 91, 274,
137, 412, 206, 103, 310, 155, 466, 233, 700, 350,
175, 526, 263, 790, 395, 1186, 593, 1780, 890,
445, 1336, 668, 334, 167, 502, 251, 754, 377,
1132, 566, 283, 850, 425, 1276, 638, 319, 958,
479, 1438, 719, 2158, 1079, 3238, 1619, 4858,
2429, 7288, 3644, 1822, 911, 2734, 1367, 4102,
2051, 6154, 3077, 9232, 4616, 2308, 1154, 577,
1732, 866, 433, 1300, 650, 325, 976, 488, 244,
122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80,
40, 20, 10, 5, 16, 8, 4, 2, 1 }
The Collatz conjecture is the
process that eventually every
number will reach one. If the
conjecture is false it can only
be because there exists a
starting number which gives
rise to a sequence which does
not contain 1. This sequence
might enter a repeating cycles
that excludes one, but no
such sequence has been
found.
Hailstone Pseudocode
• function hailstone(n)
– while n > 1
• show n
• if n is odd then
– set n = 3n + 1
• else
– set n = n / 2
• endif
– endwhile
– show n
Many numbers have
been put into the
computer program,
and every one has
reached the number
one.
But what is the Collatz problem?
The Collatz conjecture has not been proved nor
disproved
Every number that has been tried has reached the
number one
Nobody has been able to prove the Collatz
conjecture
Although the conjecture has not been proved most
mathematicians believe that it is true, because so
many numbers have been tried. In fact every
number up to 5.764 x 1018
Collatz Conjecture Proofs
Indirectly
Reverse method: Instead of proving that every
natural number leads to one. This method proves
that one leads to every natural number
Parity Sequence:
This can be done because when ever n is odd 3n=1
is even.