# Generalization of primitive roots

The standard definition is that $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the order of $a$ modulo $p$ is $p-1$.

Let me rephrase, to motivate my generalization: $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the linear recurrence defined by $x_0=1$, $x_n=ax_{n-1}$ for $n\geq1$ has maximum possible period in $\mathbb{Z}/p\mathbb{Z}$.

So, we could define $(a_1,\ldots,a_r)\in\mathbb{Z}^r$ to be an $r$-primitive root modulo $p$ when the order $r$ linear recurrence defined by $x_0=\cdots=x_{r-2}=0$, $x_{r-1}=1$, and $x_n=a_1x_{n-1}+\cdots+a_rx_{n-r}$ for $n\geq r$ has the maximum possible period (for such sequences) in $\mathbb{Z}/p\mathbb{Z}$.

Is anything known about the generalized version of primitive roots I've described? I chose my initial values $x_0=\cdots=x_{r-2}=0$, $x_{r-1}=1$ in analogy with the Fibonacci numbers, but is there a standard / most general choice? The starting values affect the period a lot, so it's important to be working with the "right" ones, I guess. What is the maximum achievable period for a linear recurrence in $\mathbb{Z}/p\mathbb{Z}$? An obvious upper bound is $p^r-1$, but it's not clear to me that this is the correct number to replace $p-1$ in the standard definition of primitive root. Is anything known in the direction of Artin's conjecture for "$r$-primitive roots", as I've called them?

Other thoughts: because there is no "formula" for the order of $a$ modulo $p$ (this would amount to a formula for the discrete logarithm), there certainly isn't a formula for the period of the linear recurrence defined by $(a_1,\ldots,a_r)$ modulo $p$. I tried to come up with one for a while before I realized this, so I just wanted to save anyone else the trouble.

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The characterization that Voloch gave implies the $r$-primitive root property is independent of the choice of nonzero initial vector $(x\_0, \dots, x\_{r-1})$, since all nonzero vectors are reached by a maximal length cycle. More specifically, if there were a shorter cycle with a nonzero initial value, then there wouldn't be enough elements left in a field with $p^r$ elements to make a disjoint cycle that satisfied the $r$-primitive root property. –  S. Carnahan Jan 12 '10 at 23:29
Ah, I see! Thanks for the explanation. –  Zev Chonoles Jan 13 '10 at 23:30

If $(a_1,\ldots,a_r)$ is an $r$-primitive root in your definition, then the polynomial $x^r-a_1x^{r-1}-\cdots-a_r$ is irreducible in $Z/pZ[x]$ and any of its roots is a generator (i.e. a primitive root) of the multiplicative group of the field of $p^r$ elements and conversely. This maximal period is $p^r-1$. See, e.g., Lidl-Niederreiter Finite Fields.

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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. –  Zev Chonoles Jan 11 '10 at 4:38
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. –  Zev Chonoles Jan 11 '10 at 4:39
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. –  Zev Chonoles Jan 11 '10 at 4:43

The proof is short enough to give here.

Proposition: A finite subgroup of the multiplicative group of a field is cyclic.

Proof. Let $G$ be a finite subgroup of the multiplicative group of a field $K$. Since $g^{|G|} = 1$ for all $g$ and since polynomials over a field have at most as many roots as their degree, it follows necessarily that $\prod_g (x - g) = x^{|G|} - 1$. By inspecting the roots of the polynomials $x^d - 1, d | |G|$, it follows that $G$ has $\phi(d)$ elements of order $d$, and this condition is equivalent to $G$ being cyclic (indeed it implies that there is at least one element of order $|G|$).

Corollary: $(\mathbb{F}\_{p^n})^{*}$ has an element of order $p^n - 1$. (This gives the primitive element theorem for finite fields.)

Let $\alpha$ be such an element and let $f(x) = x^n - a_{n-1} x^{n-1} - ... - a_0 = \prod_{i=0}^{n-1} (x - \alpha^{p^i})$ be its minimal polynomial. The generating function $\frac{1}{x^n f \left( \frac{1}{x} \right)} \bmod p$ has a partial fraction decomposition containing terms corresponding to $(p^n - 1)^{th}$ roots of unity, so its coefficients necessarily have period exactly $p^n - 1$.

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There are good results related to what you are asking. Hans Roskam stated and proved a quadratic analogue of Artin's conjecture (and I think his paper is much easier to read than Hooley's). He studies the density of primes in which a fundamental unit in a real quadratic field has a maximal order.

The paper is titled, "A quadratic Analogue of Artin's Conjecture on Primitive Roots" Journal of Number Theory 81, 93-109.

You can find his papers here and this is also good.

Hope this helps.

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