So, let's say we want to disprove Collatz. That means we're in the hunt for an illusive unicorn the first (lowest) starting number that either climbs forever or rises and falls back to itself in a loop.

Let's define these kinds of starting numbers:

• S1 is the starting number of any Collatz sequence (regular finite one that ends in 1 or a unicorn)
• I1 (Infinite) is the lowest "unicorn" number that climbs forever
• L1 (Looper) is the lowest "unicorn" number that begins a loop

What do we know about I1 or L1?

1 The sequence/path/trajectory of I1 or L1 can never fall below the start I1 or L1

First, we know (because we said they were "lowest") that they their sequences can never fall below their start (I1 or S1). But, you say, what if I1 drops below itself before rising forever? Then, there's a true I1 that's the lowest number that rises forever. Likewise, if we found an L1 that loops but that loop contains a lower number L2, that number is the true lowest number in the loop.

2 I1 or L1 must be odd (2x+1)

This follows from #1. If I1 or L2 were even the first step in their sequence would be I1/2 or L1/2 which drop below their start.

3 S1 or I1 can be a multiple of 3 but L1 cannot

This requires our first proof:

Let's say some number N is a multiple of 3 (N = 3x) and has a predecessor P.
If P is even (call it Pe), then Pe/2 = N = 3x
If P is odd (call it Po), then 3Po+1 = N = 3x
Well, Pe must be greater/higher than N and is also a multiple of 3 (23x = 3(2x)). So, an even predecessor can only be higher.
How about a lower predecessor such that 3Po+1 = 3x?
If we divide both sides by 3 we get (3*Po+1)/3 = 3x/3
Or Po+1/3 = x which can never have a solution for integer x.
Therefore, no N = 3x can have a lower predecessor in a Collatz sequence.

This is a big insight this means that no number in a Collatz sequence can be a multiple of 3 EXCEPT the start S1.

Definitions:

• S1 = Start of any Collatz sequence
• I1 = Lowest start of an infinitely rising (infinite) sequence
• L1 = Lowest start of an looping sequence that returns to L1
• M = Any number in the middle of a Collatz sequence after the start
• P = The highest (peak) number reached by a Collatz sequence
• D = The first number that drops below the start of a finite Collatz sequence
• Pred(N) = Predecessor of N is the number (or numbers) that immediately precede N in a sequence
• Succ(N) = Successor of N is the number that follows N in the Collatz sequence

Next:

• Peaks
  • Must be even
  • Must have lower predecessor
• Before Drop
  • Even number < 2 x Start
• Predecessors
  • Every number has infinitely many higher EVEN predecessors (2N, 4N, 8N, etc)
  • Every number 3x+2 has a lower ODD predecessor 2x+1 because 2x+1 -> 6x+4 -> 3x+2