User:Frostburn/Collatz scales: Difference between revisions
→Additional results: Make additional conjectures. |
Give (seemingly) unconditional form of the Collatz iteration. |
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The [[Wikipedia:Collatz conjecture|Collatz conjecture]] states that a certain sequence always reaches one regardless of the starting value. | The [[Wikipedia:Collatz conjecture|Collatz conjecture]] states that a certain sequence always reaches one regardless of the starting value. | ||
== Definition == | |||
We can formulate the Collatz iteration musically as follows: | We can formulate the Collatz iteration musically as follows: | ||
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We call the set <math>\left\{ s_n | n \in \mathbb{N} \right\} \cup \left\{ 2 \right\}</math> the '''Collatz scale''' of <math>s_0</math>. | We call the set <math>\left\{ s_n | n \in \mathbb{N} \right\} \cup \left\{ 2 \right\}</math> the '''Collatz scale''' of <math>s_0</math>. | ||
It might seem that we need to somehow compute and access the components of the reduced form, but we can make the iteration unconditional by generalizing gcd and lcm to fractions. | |||
Now the recursion is simply. | |||
<math> | |||
s_{n+1} = \bigl ( (1 + 3\ \mathrm{lcm}(1, s_n)) \cdot \mathrm{gcd}(1, s_n) \bigr ) \mathrm{red}\ 2 | |||
</math> | |||
== Conjecture == | == Conjecture == | ||