Collatz Conjecture

The Collatz Conjecture is an open unsolved mathematical problem that has currently defied the attempts of the most brilliant mathematicians to solve it.

What is the Collatz Conjecture?

While solving the Collatz Conjecture has shown itself to be next to impossible, understanding the conjecture is easy. The conjecture assumes that any number (given as X) will eventually reach 1 by using the two following rules:

If X is even (X=0 MOD 2) then divide X by 2.

If X is odd (X=1 MOD 2) then triple X and add 1.

Let's start with the number 7. Since it is odd, we use (3x+1) to get 22. 22 is divisible by 2 so we divide it by 2 and get 11. We continue onward with this and get the following numbers:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

There are several ways one can go about solving or proving a conjecture. We must prove that all numbers (X) WILL eventually reach 1 given this function. However, we could also set out proving that some number (X) cannot reach 1 given this function.

There are also certain ways to find shortcuts to a proof. For instance, if we can prove that X will eventually fall to a number less than itself, we can assume that the conjecture is true. Why? If we start with X and continue testing increasingly larger values of X (i.e. X+1) AND we also can prove that X eventually falls to a number smaller than itself, then we already know that smaller number eventually will fall to 1 since it has previously been tested.

The premise behind the conjecture assumes something very basic. Pay close attention, because this next statement is profound. The Collatz Conjecture assumes that this function for any X will eventually hit A POWER OF 2.

Huh?

Ok, let's look at the function again.

If X is even: {X / 2}
If X is odd : {3X + 1}

Pay close attention to the first part of the function. If X is even, we must divide it by 2. If, however, we begin with any X such that X = (2^Y) then X will instantly fall to 1.

Observe:

X = 64

64,32,16,8,4,2,1

X = 1024

1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1

In essense, the road home for any X in the Collatz Conjecture is for the function to eventually arrive at some power of 2. We have just proven the Collatz Conjecture for a subset of all natural numbers -- namely, all X such that X = 2^Y for any Y such that Y is a natural number.

How big is this subset? Well, since Y can be any natural number. Since there are an infinite number of them, we've just proven the Collatz Conjecture for an infinite amount of numbers but that infinity is just a sub-set of the larger infinity (namely, all natural numbers). Woah. Infinity comes in different sizes.

Since it is obvious that if any X arrives at (2^Y) [such that Y is a natural number], then X falls to 1.

Collatz Conjecture -- The Ups and Downs

Some numbers that are small take many iterations until they get to 1. Take a look at 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

27 took 111 steps and got as high as 9,232 before reaching 1. Notice also that every number in this sequence has now been proven to reach 1. Pick any number in the sequence and test it. Actually, you don't have to -- just follow the next number in the sequence.

Let's look at the function again more carefully:

X is odd: {3X + 1}
X is even: {X / 2 }

For increasingly larger values of X, the "+1" in {3X+1} becomes irrelevent. Let me make a strong distinction here, though. The +1 is relevent because after applying 3X + 1 to an odd X, the following number IS ALWAYS EVEN. The question, however, is what KIND of even number 3X+1 will fall on. It is definitely some number that is a multiple of

Take for example the number 100. Let's assume 100 was reached by using the function 3X+1 on the number 33. 100 becomes 50 and then 50 becomes 25. Since 100 is the same as (25 * 2^2), we can call 100 a "even2" number. (I just made that up and will make this more mathematically correct when I get the chance, but I am just trying to get some thoughts out quickly.

Alright, now back to the the heart of the matter.

If X is an even number that falls directly to an odd (What I'll term an "even1" number), then we know that the next number will be approximately 1.5x larger than the original X.

Observe:

X = 31

31 -> 94 (even1 number) -> 47

47 is approximately 1.5x larger than 33.


What if, however, X is a number such that 3X+1 results in an even2 natural number? We will, in essence, be increasing X by 3 and then dividing it by 4. This results in a number that is .75X.

Observe:

X = 33

33 -> 100 (even2 number) -> 50 -> 25

25 is approximately 75% of 33.


Alright, let's forget about the Colletz Conjecture for a moment and pretend that all even numbers are just even1 numbers (not true, but this is just pretend). If this were the case, the Collatz Conjecture would be false. Why? Because every time we increased X by 3, we would then decrease by dividing by 2. The numbers would constantly get larger and larger by 150%.

Observe:

X(1) = 20
X(2) = 30
X(3) = 45
etc ...

Obviously this would never get to 1. The conjecture would be proven false easily.

However, let's assume all even numbers were even2 numbers. In this case, each time we increased X by 3, we would then divide it by 4. The result would be an X that eventually decreases to 1 because each consecutive X would be 3/4'ths the previous X.

Observe:

X(1) = 16
X(2) = 12
X(3) = 9
etc ...

Now, let's pretend that half of the even numbers are even1 and the other half are even2 and, statistically speaking, X has an equal chance of falling on either. Over a large trial, we would see X moving up by 150% and subsequently decreasing to 75% of X. Would this series diverge?

X -> 1.5X -> 1.125X

Yes, this series would diverge to infinity because the net result between a large number of even1 and even2 operations would yield an X such that the next average X would be 1.125X

How many even1 numbers are there, though?

Well, let's look at even numbers more closely.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36

2 = even1
4 = even2
6 = even1
8 = even3!!!

Woah. Wait. Even3? Yes, 8 is an even3 number. Actually, 8 is a number represented by the function X*(2^3). That's really what an even3 number is. An even2 number is X*(2^2) and an even1 number is X*(2^1).

Alright, enough of the evenX talk. Let's use real mathematics:

...2= 1(2^1)
...4= 1(2^2)
...6= 3(2^1)
...8= 1(2^3)
..10= 5(2^1)
..12= 3(2^2)
..14= 7(2^1)
..16= 1(2^4) [even4 number!]
..18= 9(2^1)
..20= 5(2^4)
..22=11(2^1)
..24= 3(2^3)
..26=13(2^1)
..28= 7(2^2)
..30=15(2^1)
..32= 1(2^5) [even5 number!]




(to be continued ...)


Scratch Pad Stuff:

(3X+1)/2 for all odd X.

Half of all even numbers are X(2^1) where X is odd.
2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, etc.

One Quarter of all even numbers are X(2^2) where X is odd.
4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, etc.

One Eighth of all even numbers are X(2^3) where X is odd.
8, 24, 40, 56, 72, 88, 104, 120, 136, 152, 178, 194, 210, etc.

Understanding Space-Time

Understanding Space-Time Part I:
What Exactly *is* the Speed of Light?

One of the hardest concepts to grasp about space-time theory is that time is relative to the observer and is not absolute. The universe as a whole does not follow some "master clock" that encompasses everything.

The speed of light has been measured using various means with significant accuracy. That speed is 186,282 miles per second in a vaccuum. However, that speed is relative to a stationary observer on Earth. How fast would the speed of light appear to be if you trailed behind it at 99% of its speed? It would still appear to leave you at the same speed.

Take another example using the classic unit of measurement to find the distance to a star -- the light year. Suppose that a star was said to be 15 light-years away from Earth. What exactly does that mean? Well, it could mean several things depending on your frame of reference. To the observer on Earth, if a spaceship were to leave and head towards this star at the speed of light, the observer on Earth would notice 15 years later that the spaceship finally made it to that star. What about from the perspective of someone on that ship?

Well, the person on that ship would notice that the trip was instantaneous. How is this possible? How could the ship take both 15 years and, at the same time, get their instantaneously?

The answer lies in something called "relativity."

To be continued ...