Timeline for Generalization of primitive roots
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 13, 2010 at 3:55 | vote | accept | Zev Chonoles | ||
| Jan 11, 2010 at 4:43 | comment | added | Zev Chonoles | Of course, we could always reformulate the question to just be about how an element of $\mathbb{Z}[X]$ factors in each $\mathbb{Z}/p\mathbb{Z}[X]$, and ignore the relationship with sequences - which I'm not entirely against, since it seems like an intriguing question all on its own. | |
| Jan 11, 2010 at 4:39 | comment | added | Zev Chonoles | Furthermore, even if the above conditions are met and $X^r-a_1X^{r-1}-\cdots-a_r$ is irreducible in $\mathbb{Z}/p\mathbb{Z}[X]$, it seems possible that $x_n=c_1\lambda_1^n+\cdots+c_n\lambda_r^n$ could repeat earlier than the period of the $\lambda_i$ individually. This is why I was concerned about the apparent lack of "canonical" starting values: the $c_i$ could cause earlier periodicity than the roots of the polynomial "warrant", in some sense. | |
| Jan 11, 2010 at 4:38 | comment | added | Zev Chonoles | Thanks for your answer and reference! I still have some concerns, so I'd appreciate it if you could explain further. For example, I seem to remember that we can only write $x_n=c_1\lambda_1^n+\cdots+c_n\lambda_r^n$, where the $\lambda_i$ are the roots of $X^r-a_1X^{r-1}-\cdots-a_r$ and the $c_i$ are determined by the starting values, if the $\lambda_i$ are in fact all distinct, and $r<p$, so I was worried that some sort of degenerate case could mess any nice characterization up. | |
| Jan 11, 2010 at 0:03 | history | answered | Felipe Voloch | CC BY-SA 2.5 |