Forgive my lack of understanding if this is stupid

>>7761135
Reformulate your question. Makes no sense to me at the moment.

>returns original number through same operation

x+0=x
X*1=x

>>7761148
I will try. So lets say we have the following:
A, B, C, D, E, F, G, H
And A=H.
This is a sort of hypothetical and abstract representation of what I'm looking for.
I have been reading about recursion a little bit, but I'm not sure if that's precisely what I mean.

>>7761168
[math]
F_n mod(x)
[/math]

>>7761135
> (and this sequence should be in a number system which uses a real number for a base, for example base-φ)
That doesn't matter at all. If the sequence does something like being periodic in one base, it'll do it in any other base too.

So, you're probably looking for recursively defined periodic sequences. There are many ways to find some of them. The simplest non-trivial one would probably be the one where you start from any value for u_0, and you define u_{n+1}=-u_n. That way, the sequence can be for instance 4, -4, 4, -4, 4, -4 etc.

If you want a given specific periodicity n instead of periodicity 2, adapt the above method of multiplying by -1, and instead multiply with a primitive n-th root of unity at each step, so that in the end you go back to 1. If you don't want to use complex numbers, doubly-recursive sequences work really well, e.g., u_{n+2}=-u_{n+1}-u_{n}. If you don't want those either, you can just use polynomial interpolation to make a function that does what you want, for instance have a sequence that loops like 0,3,2,5,8,0,3,2,5,8,0,3,2,5,8,...

If you don't want that either, then take some time and ask the question better.

Maybe this will make more sense. I mentioned φ-base representation earlier. φ is 1.6180... while the conjugate root of φ is -0.618...
Now I was never fantastic at math, but those can both be represented with the symbol φ, right? so could you hypothetically, through addition, have a sequence of addition (with positive numbers) which both begins with φ and ends with φ.

>>7761187
That doesn't matter at all. If the sequence does something like being periodic in one base, it'll do it in any other base too.
It matters. I'm doing research for a novel and this is thematically very relevant. Irrational numbers have historically been said to be "close to the divine."

>>7761240
Fuck off faggot

>>7761251
That's about what I expected. I did get some good answers in this thread though

>>7761240
Dude, you don't even know enough about math to ask the question properly, I give you an educated answer, telling you that the property you're looking for is going to be present in a sequence regardless of the base you write it in because it's an inherent property of the numbers and doesn't depend on how you write them down, and you go thinking that your pitiful prejudices are more likely to be correct than my 6 years in academia?

It's like saying that if you translate "one plus one equals two" in French, it might not be true anymore. Bases are a way to write numbers. The property you're looking for is a property of numbers, if you want to write them in a redneck accent base, they still mean the same fucking thing, you dumb fuck.

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>>7761770