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The Collatz Conjecture
Let's write it in the style of the old Basic programming language: 10 Pick any positive integer n. For the first digits, results are:
The site pass.maths.org.uk says that "These are sometimes called 'Hailstone sequences' because they go up and down just like a hailstone in a cloud before crashing to Earth - the endless cycle 4, 2, 1, 4, 2, 1. It seems from experiment that such a sequence will always eventually end in this repeating cycle 4, 2, 1, 4, 2, 1,... and so on, but some values for N generate many values before the repeating cycle begins. For example, try starting with n = 27. See if you can find starting values that generate even longer sequences". And it puts a slot so you can enter your chosen number (27, in this case) to be investigated, and the resulting chain of integers is given in a neat presentation: Hailstone sequence 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 So asks Lagarias: is the 3x+1 problem intractably hard? He states that the difficulty of settling the 3x+1 problem seems connected to the fact that it is a deterministic process that simulates "random" behavior. We face this dilemma: On the one hand, to the extent that the problem has structure, we can analyze it - yet it is precisely this structure that seems to prevent us from proving that it behaves "randomly". On the other hand, to the extent that the problem is structureless and "random", we have nothing to analyze and consequently cannot rigorously prove anything. Of course there remains the possibility that someone will find some hidden regularity in the 3x+1 problem that allows some of the conjectures about it to be settled. "The existing general methods in number theory and ergodic theory do not seem to touch the 3x+1 problem; in this sense it seems intractable at present. Indeed all the conjectures made in this paper seem currently to be out of reach if they are true; I think there is more chance of disproving those that are false. If the 3x+1 problem is intractable, why should one bother to study it? One answer is provided by the following aphorism: 'No problem is so intractable that something interesting cannot be said about it'. Study of the 3x+1 problem has uncovered a number of interesting phenomena; I believe further study of it may be rewarded by the discovery of other new phenomena. It also serves as a benchmark to measure the progress of general mathematical theories", asserts Lagarias. One of the greatest numbers subjected to the Collatz conjecture that had its algorithm subjected to computerized calculation was described by its author in Oliveira e Silva, T. "Maximum Excursion and Stopping Time Record-Holders for the 3x+1 Problem: Computational Results". Math. Comput. 68, 371-384, 1999: 3*2^53, which can also be stated as 2702*10^16, according to http://mathworld.wolfram.com/CollatzProblem.html. Related news:
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That's
the Collatz conjecture, also known as the 3n+1 conjecture, the Ulam
conjecture or the Hailstone sequence, to which names Jeffrey C.
Lagarias, which also discusses more deeply its implications at 
