# Dialogue 10 – OddCore on the Koch Curve

**Speaker: Hello Nara!**

**We’ve come so far that I want to know even more. In the cells of the table from last time there are also even numbers. But I’m not interested in them. Please divide all values by 2 until they are odd.**

Nara: You’ve got the taste now and the Collatz fever has gripped you!

What you mean are the so‑called odd‑core numbers – that is, the odd kernel of even numbers. Mathematically, one uses the so‑called 2‑adic valuation, which leads us to the dyadic representation. In this approach we focus only on the essential core of the number.

**Speaker: Oh God! What’s that?**

Nara: It sounds more complicated than it is! Dyadic numbers are simply binary numbers, which we have already worked with extensively. Roughly speaking, we just cut off all the trailing zeros at the end of the binary number. The core then begins with a 1 and ends with a 1. There is also an entry in the OEIS under the identifier A000265.

Take a look.

**Speaker: It says something about a fractal sequence. What’s that about?**

Nara: Fractals are geometric figures that are formed from very simple rules. You can repeat them infinitely often, and with each magnification you discover the same basic pattern again. Maybe you know the Koch curve? That’s a classic example.

**Speaker: No, that doesn’t ring a bell. Please explain it more clearly.**

Nara: Imagine a straight line. We divide it into three equal parts. We remove the middle part and replace it with two sides of an equilateral triangle – like a little “mountain” in the middle.

That’s the first step. And now comes the fractal: we take each of these new, smaller segments and repeat exactly the same rule.

A structure emerges that becomes more and more complex, but at its core it always follows this one simple instruction.

**Speaker: I see! The pattern “grows” out of itself. And what does that have to do with our odd‑core numbers?**

Nara: We take a Koch curve of depth 1. Depth means how many times we repeat the process. And then we write a natural number at each corner point, starting with 2: 2, 3, 4, 5, 6.

And now we divide all even numbers by 2 until they yield an odd number. That gives 1, 3, 1, 5, 3.

Then we don’t start with 2, but with 4: 4, 5, 6, 7, 8.

For these numbers we again generate the odd core: 1, 5, 3, 7, 1.

**Speaker: OK, and what’s the point? What happens if we go deeper?**

Nara: Exactly! We now increase the depth of the Koch curve to 2. Such a curve has exactly 16 elements. And this time we start with 8.

**Speaker: Wait, with 16 elements only the numbers from 8 to 24 fit!**

Nara: Good observation! But look at what’s at the top: 16. And because that’s a power of two, it reduces to 1 – its core.

**Speaker: OK… and what if next time we start with 16?**

Nara: Then our interval goes from 16 to 32. In the middle is 24. And its odd core is 3.

**Speaker: Uh, and what’s the secret behind this?**

Nara: There is no secret! A Koch curve of depth k always has exactly 4 to the power of k segments and 4 to the power of k + 1 points. If we start our interval exactly at such a power – that is, at 4, 16, or 64 – then in the middle there is always an even number whose odd core is 3.

**Speaker: Well, and if we start offset?**

Nara: If we start with powers of two that have an odd exponent – that is, 21, 23, 25, and so on – then at the top there is always a power of two with an even exponent – that is, 22, 24, 26, and so on. And their core is 1. That is the fractal property of odd‑core numbers: the top of the curve oscillates between 1 and 3.

**Speaker: Phenomenal! But what does this have to do with the Collatz conjecture?**

Nara: You’re right. This doesn’t help us with the Collatz conjecture for now. But we understand the relationship between even and odd numbers better.

Would you like to know more connections?

**Speaker: No, thanks! Somehow I’m still frustrated that we’re not making progress.**

Nara: Patience! You’re on an excellent path! If you like, you can try out what we discussed yourself. The link is in the video description.

Source: [https://www.3n-1.de/cms/en:collatz:10](https://www.3n-1.de/cms/en:collatz:10)