Blog post: Unity+: Patterns in Collatz Hailstone Sequences - Collatz Numbers

As mathematicians try to solve the Collatz conjecture, there have mainly been methods of analysis dealing with the numbers in a Hailstone sequence produced by the functions of the Collatz conjecture, which are 3x+1 and x/2(the problem defines them as 3n+1 and n/2, but for Collatz Theory it is better for them to be known as 3x+1 and x/2). Many attempts to solve the problem have dealt with the patterns of iterations in the Hailstone sequence.

However, there is not definable pattern within the iterations, though there seems to be one visible. Though this is one possible way to solve the Collatz conjecture, I believe there is another way to do so. Instead of observing the iterations, maybe the numbers themselves should be observed. For example, here is a hailstone sequence for the number 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 }

Though, not visible at first, here is what is a definable pattern within the Hailstone sequence and in many other Hailstone sequences.

{ 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 }

All the numbers in bold are what are known as Collatz Numbers. A Collatz Number is defined by the numerator of the derivative of the product of a function and its inverse.

All these numbers are Collatz numbers. However, there are spaces where instead of a one space difference here is, in fact, a two-space difference. These areas in the Hailstone sequence are known as Hailstone Exceptions, or Hailstone remainders. These are the result of a Collatz number that is multiplied by two. In this case, the equation for finding a Collatz Number is 6x-2.

Now, since Collatz numbers will always reach the value of one using the original function and the derivative value of r all that would have to be proven is that this pattern would always exist for all natural numbers. It seems like a potential solution, but it just seems to simple to be a solution to a problem that has lasted 76 years unsolved. However, it may give insight into the actual solution to the problem.

In the next post on this blog, I will bring up what is called Raymond Arithmetic. Thank you for reading.

Sources:

"Collatz Conjecture." Wikipedia. Wikimedia Foundation, 29 Nov. 2013. Web. 30 Nov. 2013. .
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