The Collatz Conjecture

This is also known as the ‘half or triple plus one’ problem.

Let $$n$$ be any arbitrary positive integer.

If $$n$$ is even, then divide it by two.

Otherwise, if $$n$$ is odd, then multiply it by three and add one.

We have the piecewise function :

$$f(n) = \begin{cases} n/2, & \text{if n is even}\\ 3n + 1, & \text{if n is odd} \end{cases}$$

Now iterate these steps to form a sequence.

The Collatz conjecture states that the sequence will eventually reach the number one for all $$n$$.

Like many open conjectures in arithmetic (number theory) it is easily understood but very difficult to prove.

The first ten sequences :

$$n = 1 : 1, 4, 2, 1.\\ n = 2 : 2, 1.\\ n = 3 : 3, 10, 5, 16, 8, 4, 2, 1.\\ n = 4 : 4, 2, 1.\\ n = 5 : 5, 16, 8, 4, 2, 1.\\ n = 6 : 6, 3, 10, 5, 16, 8, 4, 2, 1.\\ n = 7 : 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.\\ n = 8 : 8, 4, 2, 1.\\ n = 9 : 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.\\ n = 10 : 10, 5, 16, 8, 4, 2, 1.\\$$

These sequences are often referred to as the hailstone numbers because they rise and fall before eventually dropping down to one.

The sequence for $$n = 27$$ takes $$111$$ steps rising to $$9,232$$ before falling to $$1$$.

All positive integers up to $$2^{64}$$  have been tested and no counter-example has been found, however, this does not constitute a proof.

We can look at the conjecture extended over the complex plane and see the fractal generated by the iterations.  The Collatz function over the real numbers is defined as :

$$f(x) = \bigl(1 + (-1)^x\bigr) \frac{x}{4} + \bigl(1 \, – (-1)^x \bigr) \frac{3x +1}{2}$$

Over the complex plane it is defined as :

$$f(z) = \frac{1}{4} \bigl(2 + 7z \, – (2 + 5z)\cos(\pi z) \bigr)$$

since $$(-1)^x = \cos(nx)$$.

The Collatz $$f(z)$$ fractal :

The large black region is around $$z = 0$$.  The spikes to the left and right are $$z = -2, -1, 1$$ and $$1$$.

Nathaniel Johnston introduces a modified Collatz function :

$$g(z) = \frac{z}{4} \bigl(1 + \cos(\pi z) \bigr) + \bigl(\frac{3z + 1}{16} \bigr) \bigl(1 \, – \cos(\pi z) \bigr) \bigl(3 \,- \sqrt2\cos \bigl((2z \, -1)\frac{\pi}{4} \bigr) \bigr)$$

His reason for doing so being that $$g(1) = 1$$.  That is, $$1$$ is a fixed point of $$g(z)$$.

The Collatz $$g(z)$$ fractal :

Close-up of the $$g(z)$$ fractal at $$z = 1$$ :

$$g(z)$$ at $$z = 4$$ :

$$g(z)$$ at $$z = 8$$ :

$$g(z)$$ at $$z = 16$$ :

$$g(z)$$  at $$z = -8$$ :

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