So, forgive my ignorance, this isn't a troll just a question.

How are continued fractions related to modular arithmetic?

Take the following example:

70%32 = 6
70//32 = 2

which we can just represent as
6+(32*2)

Now, as a continued fraction, we would do

70/32 = 2 + 6/32
and repeating what we just did but for 6/32, as you would with a continued fraction.

Am I actually doing everything correctly here or am I missing something?

It should be possible to prove the collatz conjecture by mapping the unit digit transitions between numbers, namely into a finite state machine. From there we could use predicates and quanitifiers to prove, by process of exclusion, that for any given combination of 10s digit and 1s digit, no number can transition to anything but whats specified in the state machine assuming that number equals x in x3+1 or x/2

Ipso facto, a series of equations proving by process of elimination, that state machines transitions are the only allowable ones, would prove the collatz conjecture by proving the fsm is a valid representation for any given integer N.

I'm actually working on it now but I don't know enough about modular arithmetic and predicate logic to write a proof. I just have the state diagrams on some dot matrix paper at the moment.

If anyone wants to beat me to it, feel free.

So for example any number ending in 13, will, after x3+1, end in 40.

Any number ending in 40 will end in 20. Any number ending in 20 will end in 10, which will end in 5 as the unit digit.

It's easier to prove in the single digit case, and the finite state machine for that is already written, at least on paper.

I'll post pictures when I get a chance.

Brain fart.

In Java and many other languages there are basic types, like char and String. So why does Java have char and String, but not a digit type?

A number is basically a series of digits. For modular arithmetic it is very useful to be able to extract the 3 in the number 1234, it's just the 3rd digit in a number.

Base 2, base 10, base anything could be supported easily too. E.g. a base 2 digit would be:

digit d = 0b2; // or 1b2, but 2b2 would be a compilation error

A number would then be some kind of string of digits.

Any thoughts on this?