It seems that a lot of work is being done in trying to come up with proofs for the Collatz Conjecture.

I have been playing around myself with some equations and may have come up with a new method to help determine a proof, but am unsure if what I have come up with is just as complicated or does in fact have any practicality...

I started by looking at the Collatz Conjecture for positive integers.

For all positive integers x, where x>=1, the next value is found by the following formula

F(x) = 3x+1 ; when x is odd

F(x) = x/2 ; when x is even

Looking at just a single step of the function, we see that the following occurs:

1 -> 4 ; 2 -> 1 ; 3 -> 10 ; 4 -> 2 ; 5 -> 16 ; 6 -> 3 ; 7 -> 22 ; 8 -> 4 etc.

Looking for patterns, I noticed something very basic, which seems to have been overlooked.

Let us take our original set of input values and split them up into the following subsets:

A(x) = 4x : x all positive integer where x>=1

B(x) = 4x - 2 : x all positive integer where x>=1

C(x) = 4x - 1 : x all positive integer where x>=1

D(x) = 4x - 3 : x all positive integer where x>=1

If we now relook at the single step from the function F(x) we get this:

D(1) -> A(1) ; B(1) -> D(1) ; C(1) -> B(3) ; A(1) -> B(1) ; D(2) -> A(4) ; B(2) -> C(1) ; C(2) -> B(6) ; A(2) -> A(1) ; etc.

In fact, we get the generalized change as follows

F(A(x)) -> A(x/2) : for all positive integer x where x is even

F(A(x)) -> B((x+1)/2) : for all positive integer x where x is odd

F(B(x)) -> C(x/2) : for all positive integer x where x is even

F(B(x)) -> D((x+1)/2) : for all positive integer x where x is odd

F(C(x)) -> B(3x) : for all positive integer x

F(D(x)) -> A(3x-2) : for all positive integer x

We can see that all C(x) go to B(3x) and all D(x) go to A(3x-2), so can simplify the 6 equations to 4

F(A(x)) -> A(x/2) : for all positive integer x where x is even

F(A(x)) -> B((x+1)/2) : for all positive integer x where x is odd

F(B(x)) -> B(3x/2) : for all positive integer x where x is even

F(B(x)) -> A((3x-1)/2) : for all positive integer x where x is odd

These formulas can give us the next term in the sequence for any number put into the Collatz formula, and is also recursive.

Thus if we can prove that the only loop these formula have is the trivial A(1) -> B(1) -> A(1), then we show that there are no other loops possible for the Collatz conjecture.

And if we can prove that the collection of formula tend toward shrinking the value of x, we show that any positive integer will eventually reach 1 for the Collatz conjecture.

I have been playing around myself with some equations and may have come up with a new method to help determine a proof, but am unsure if what I have come up with is just as complicated or does in fact have any practicality...

I started by looking at the Collatz Conjecture for positive integers.

For all positive integers x, where x>=1, the next value is found by the following formula

F(x) = 3x+1 ; when x is odd

F(x) = x/2 ; when x is even

Looking at just a single step of the function, we see that the following occurs:

1 -> 4 ; 2 -> 1 ; 3 -> 10 ; 4 -> 2 ; 5 -> 16 ; 6 -> 3 ; 7 -> 22 ; 8 -> 4 etc.

Looking for patterns, I noticed something very basic, which seems to have been overlooked.

Let us take our original set of input values and split them up into the following subsets:

A(x) = 4x : x all positive integer where x>=1

B(x) = 4x - 2 : x all positive integer where x>=1

C(x) = 4x - 1 : x all positive integer where x>=1

D(x) = 4x - 3 : x all positive integer where x>=1

If we now relook at the single step from the function F(x) we get this:

D(1) -> A(1) ; B(1) -> D(1) ; C(1) -> B(3) ; A(1) -> B(1) ; D(2) -> A(4) ; B(2) -> C(1) ; C(2) -> B(6) ; A(2) -> A(1) ; etc.

In fact, we get the generalized change as follows

F(A(x)) -> A(x/2) : for all positive integer x where x is even

F(A(x)) -> B((x+1)/2) : for all positive integer x where x is odd

F(B(x)) -> C(x/2) : for all positive integer x where x is even

F(B(x)) -> D((x+1)/2) : for all positive integer x where x is odd

F(C(x)) -> B(3x) : for all positive integer x

F(D(x)) -> A(3x-2) : for all positive integer x

We can see that all C(x) go to B(3x) and all D(x) go to A(3x-2), so can simplify the 6 equations to 4

F(A(x)) -> A(x/2) : for all positive integer x where x is even

F(A(x)) -> B((x+1)/2) : for all positive integer x where x is odd

F(B(x)) -> B(3x/2) : for all positive integer x where x is even

F(B(x)) -> A((3x-1)/2) : for all positive integer x where x is odd

These formulas can give us the next term in the sequence for any number put into the Collatz formula, and is also recursive.

Thus if we can prove that the only loop these formula have is the trivial A(1) -> B(1) -> A(1), then we show that there are no other loops possible for the Collatz conjecture.

And if we can prove that the collection of formula tend toward shrinking the value of x, we show that any positive integer will eventually reach 1 for the Collatz conjecture.

Last edited: