I have a short list of favorite unsolved problems in Number Theory. I'll start with the Hailstone numbers, aka the Hailstone sequences:

Starting with any positive integer n, form a sequence in the following way

>> If n is even, divide it by 2 to give n' = n/2.

>> If n is odd, multiply it by 3 and add 1 to give n' = 3n + 1.

E.g.

n = 5 ---> 16, 8, 4, 2, 1, 4, 2, 1, ...

n = 11 --> 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...

n = 13 --> 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...

n = 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, 4, 2, 1, ...

n = 100 ---> 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...

Can it be proved that every starting value will generate a sequence that eventually settles to 4, 2, 1, 4, 2, 1,...?

Try and generate more sequences with the Hailstone Evaluator

http://plus.maths.org/content/mathematical-mysteries-hailstone-sequences

This is also known as the Collatz Problem

http://mathworld.wolfram.com/CollatzProblem.html

It works for every number you try. Does it mean that it would work for all numbers?

No one knows for sure. Just because it works for every number you try -- big or small -- doesn't guarantee that it works for all numbers.

And, why the pattern (4, 2, 1, 4, 2, 1,) keeps popping up?

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