# Order-finding problem / period finding?

For positive integers x and N, such that x<N, with no common factors, the order of x modulo N is defined to be the least positive integer r such that:

$$x^r≡1(modN)$$

The order-finding problem is determining the order r of a specified x and N. Link: https://www.youtube.com/watch?v=YNq7x6z8F1g

My questions:

1. Is it not 1modN, always 0 (then we want an r such that $$x^r$$ is 0, this sounds weird)? Maybe I am misunderstanding the ≡ symbol.
2. Is it this problem exactly the period finding problem with another name?

Thanks!

• Would Mathematics be a better home for this question? – Jonas Apr 2 at 10:57
• @Jonas Well, it relates to quantum computing, so I'd consider it on-topic here, even though the "bare" question is of course purely mathematics. (Otherwise, we'd have to migrate lots of problems - basically any mathematical question which can be understood and answered without the physical motivation in mind, such as this one: physics.stackexchange.com/questions/626267/…) – Norbert Schuch Apr 2 at 11:08

1. No, it is 1 mod N: $$x^0=1$$, and then you want to know the smallest $$r>0$$ where $$x^r=1$$ again (modulo $$N$$). Ten, of course, also $$x^{2r}=1$$, $$x^{3r}=1$$, etc. (all modulo $$N$$), so this has a period $$r$$.
2. No. Period finding is more general. This is finding the period of a specific type of function, namely a function of the form $$f_{x,N}(r)=x^r\mod N$$.