In number theory, we always write a ≡ b (mod p) means that the reminder of division of a by p and the reminder of b by p are the same. Today, I have a question about this definition.
1. Same reminder of integers:
If , a ≡ b (mod p) means or . It's quite easy to see that.
2. Same reminder of rational numbers:
We also write , for example: . What do they mean?
The above definition is obviously wrong, since . But with the modular multiplicative inverse, can be written as a * b - 1 ≡ r (mod p). So, in this case, I think the definition should be: means: For every pair of integers (A, B) satisfies , we always have . Note that so we can use the definition for integers.
3. Same reminder of irrational numbers:
Fn ≡ 276601605(691504013n - 308495997n) (mod 109 + 9)
The correctness of the last equation can be proved easily. However, to get the final equation, we need some transform like: .
My question is, how can we define equations a ≡ b (mod p), when a or b is irretional number like that, and why the above transform is correct.
I'm looking forward to hearing your responses. Thank you for your help!
P/s: I know, this blog may have something difficult to be understood, but please do not vote down it. I have thought a lot about this problem, but I can't find the answer.