On the distinctness of primitive sequences over Z/(p ^e q) modulo 2

Abstract

Let N be an integer greater than 1 and Z/(N) the integer residue ring modulo N. Extensive experiments seem to imply that primitive sequences of order n≥2 over Z/(N) are pairwise distinct modulo 2. However, efforts to obtain a formal proof have not been successful except for the case when N is an odd prime power integer. Recent research has mainly focussed on the case of square-free odd integers with several special conditions. In this paper we study the problem over Z/(p ^e q), where p and q are two distinct odd primes, e is an integer greater than 1. We provide a sufficient condition to ensure that primitive sequences generated by a primitive polynomial over Z/(p ^e q) are pairwise distinct modulo 2.

Appendix A: proof of Proposition 1

We first briefly introduce Galois rings. The notation and definitions we will use here are from [12].

A Galois ring is a finite commutative ring R with identity 1 in which the set of all zero divisors has the form p R for some prime p. Primary examples of Galois rings are integer residue rings Z/(p ^e) and finite fields $GF\left (q\right )$ of q elements. A Galois ring R is uniquely determined up to isomorphism by its characteristic p ^e and the number of elements q ^e, where q=p ^r. Therefore in what follows we denote such a ring by $GR\left (q^{e},p^{e}\right )$. In particular, $GR\left (p^{e},p^{e}\right )=\mathbf {Z} /(p^{e})$. Let $R^{\prime }=GR\left (q^{en},p^{e}\right )$ be an extension of degree n of $R=GR\left ( q^{e},p^{e}\right )$. We denote by $\textmd {Aut}\left ( R^{\prime }/R\right )$ the set of all automorphisms of the ring $R^{\prime }$ that fix each element of R. The group $\textmd {Aut} \left (R^{\prime }/R\right )$ is a cyclic group of order n generated by some automorphism σ:

$$\textmd{Aut}\left( R^{\prime}/R\right) =<\sigma>= \left\{1,\sigma, {\ldots} ,\sigma^{n-1}\right\} \text{.} $$

Moreover, for $\alpha \in R^{\prime }$, $\ \sigma \left (\alpha \right ) =\alpha$ iff α∈R, see [14, Theorem 14.30].

If f(x) is a basic irreducible polynomial of degree n over Z/(p ^e), then all the roots of f(x) belong to $GR\left ( p^{en},p^{e}\right )$. Moreover if α is such a root in $GR\left (p^{en},p^{e}\right )$, then $\alpha ,\sigma \left (\alpha \right ) ,{\ldots } ,\sigma ^{n-1}\left (\alpha \right )$ are all the roots of f(x) in $GR\left (p^{en},p^{e}\right )$, where σ is the generator of the cyclic group $\textmd {Aut}\left (GR(p^{en},p^{e})/GR(p^{e},p^{e})\right )$.

To prove Proposition 1, we need the following four lemmas.

Lemma 6

([15, Theorem 2]) Let p ^e be a prime power and $f\left (x\right )$ a monic polynomial of degree n over Z/(p ^e) that has no multiple factors over Z/(p). Suppose α ₀ ,α ₁ ,…,α _n−1 are all roots of $f\left (x\right )$ in $GR\left (p^{em},p^{e}\right )$ for some integer m. Then for any $\underline {a}=(a(t))_{t\geq 0}\in G(f(x),p^{e})$ , there uniquely exists $\beta _{0},\beta _{1},{\ldots } ,\beta _{n-1}\in GR\left (p^{em},p^{e}\right )$ such that

$$ a\left( t\right) =\beta_{0}{\alpha_{0}^{t}}+\beta_{1}{\alpha_{1}^{t}}+{\ldots} +\beta_{n-1}\alpha_{n-1}^{t}\text{,}t \geq 0\text{.} $$

(9)

Inversely, if a sequence $\underline {a}=(a(t))_{t\geq 0}$ over Z/(p ^e) satisfies (9), then $\underline {a}\in G(f(x),p^{e})$.

Lemma 7

([16, Proposition 6.1]) Let p be a prime number and $f\left (x\right )$ a primitive polynomial of degree n ≥ 2 over Z/(p). Let $\underline {a}\in G^{\prime }(f(x),p)$ and s a positive integer. Then the minimal polynomial of $\underline {a}^{\left (s\right ) }$ is irreducible over Z/(p) with degree dividing n and

$$per \left( \underline{a}^{\left( s\right)}\right) =\frac{p^{n}-1}{\gcd \left( p^{n}-1,s\right)}\text{.} $$

Lemma 8

([17]) Let p ^e be a prime power and $\underline {a}=(a(t))_{t\geq 0}$ a sequence over Z/(p ^e). Then the minimal polynomial g(x) of $\underline {a}$ over Z/(p ^e) is unique iff g(x) is a basic irreducible polynomial.

Lemma 9

([18, Theorem 11.1]) Let p ^e be a prime power and $f\left (x\right )$ a basic irreducible polynomial of degree n over Z/(p ^e). Suppose α is a root of $f\left (x\right )$ in $GR\left (p^{en},p^{e}\right )$ , then per(f(x),p ^e ) = $ord\left (\alpha \right )$ , where $ord\left (\alpha \right )$ is the least positive integer s such that α ^s =1.

Now we start to prove Proposition 1.

Proof

(Proof of Proposition 1) Let g(x) be a minimal polynomial of $\underline {a}^{\left (s\right )}$ over Z/(p ^e). Then it is clear that $g\left (x\right ) \left (\bmod {~p}\right )$ is a characteristic polynomial of $\left [\underline {a}^{\left (s\right )} \right ]_{\bmod {p}}$ over Z/(p). By Lemma 8, it suffices to show that g(x) is a basic irreducible polynomial polynomial of degree n over Z/(p ^e) only depending on f(x).

Let h(x) be the minimal polynomial of $\left [\underline {a}^{\left (s\right )}\right ]_{\bmod {p}}$ over Z/(p). Then it follows from Lemma 7 that h(x) is irreducible over Z/(p) with degree dividing n and

$$per \left( \left[\underline{a}^{(s)}\right]_{\bmod{~p}}\right) =\frac{p^{n}-1}{\gcd \left( p^{n}-1,s\right)}\text{.} $$

Since $\frac {p^{n}-1}{\gcd \left (p^{n}-1,s\right )}>p^{n/2}$ by assumption, it follows that $\deg h\left (x\right ) =n$(for otherwise $\deg h(x)\leq n/2$, which yields $per\left ( \left [\underline {a}^{(s)}\right ]_{\bmod {p}}\right ) =per(h(x),p)\leq p^{n/2}-1$, a contradiction). Since $g\left (x\right ) \left (\bmod {~p}\right )$ is a characteristic polynomial of $\left [ \underline {a}^{\left (s\right )}\right ]_{\bmod {p}}$ over Z/(p), we have

$$ \deg g(x)=\deg \left( g(x)\left( \bmod{~p}\right)\right)\geq \deg h(x)=n\text{.} $$

(10)

On the other hand, let $R^{\prime }=GR\left ( p^{en},p^{e}\right )$, R=Z/(p ^e), and $\textmd {Aut}\left ( R^{\prime }/R\right )=<\sigma >$. Suppose α is a root of $f\left (x\right )$ in $R^{\prime }$, then $\alpha ,\sigma \left (\alpha \right ) ,{\ldots } ,\sigma ^{n-1}\left (\alpha \right )$ are all the roots of $f\left (x\right )$ in $R^{\prime }$. By Lemma 6 there uniquely exist $\beta _{0},\beta _{1},{\ldots } ,\beta _{n-1}\in R^{\prime }$ such that

$$a\left( t\right)=\beta_{0} \alpha^{t}+\beta_{1}(\sigma \left( \alpha \right))^{t}+{\ldots} +\beta_{n-1}(\sigma^{n-1}\left( \alpha \right))^{t} ~\text{for}~t\geq 0\text{,} $$

and so

$$ a^{\left( s\right)}\left( t\right)=a(st)=\beta_{0}(\alpha^{s})^{t}+\beta_{1}(\sigma \left( \alpha^{s}\right))^{t}+{\ldots} +\beta_{n-1}(\sigma^{n-1}\left( \alpha^{s}\right))^{t}~\text{for}~t\geq 0\text{.} $$

(11)

Let k be the least positive integer such that $\sigma ^{k}\left (\alpha ^{s}\right )=\alpha ^{s}$. It is clear that k∣n and $\alpha ^{s},\sigma \left (\alpha ^{s}\right ) ,{\ldots } ,\sigma ^{k-1}\left (\alpha ^{s}\right )$ are pairwise distinct. Then (11) can be rewritten as

$$a^{\left( s\right)}\left( t\right) =\beta_{0}^{\prime}(\alpha^{s})^{t}+\beta_{1}^{\prime}(\sigma \left( \alpha^{s}\right))^{t}+{\ldots} +\beta_{n-1}^{\prime}(\sigma^{k-1}\left( \alpha^{s}\right))^{t}~\text{for}~ t\geq 0\text{,} $$

where $\beta _{i}^{\prime }=\sum \limits _{j=0}^{\left (n/k\right )-1} \beta _{jk+i}$ for 0≤i≤k−1. Set

$$m(x)=\prod\limits_{i=0}^{k-1}\left( x-\sigma^{i}\left( \alpha^{s}\right) \right) \text{.} $$

Since $\sigma \left ( m(x)\right )=m(x)$ and m(x) is a monic polynomial over Z/(p ^e), it follows from Lemma 6 that m(x) is a characteristic polynomial of $\underline {a}^{\left (s\right )}$ over Z/(p ^e), and so

$$ \deg g(x)\leq \deg m\left( x\right)=k\leq n\text{.} $$

(12)

Combining (10) and (12) we obtain that $\deg g\left (x\right )=n$. Now we have

$$\deg \left( g(x)(\bmod{~p})\right)=\deg g(x)=n=\deg h(x)\text{,} $$

it follows that both $g(x)\left ( \bmod {~p}\right )$ and h(x) are the minimal polynomial of $\left [\underline {a}^{\left ( s\right ) }\right ]_{\bmod {p}}$ over Z/(p), and so we get that $g(x)\left (\bmod {~p}\right ) =h(x)$ (since it is well-known that the minimal polynomial of a sequence over the finite field Z/(p) is unique). Thus we have showed that $g\left (x\right )$ is a basic irreducible polynomial of degree n over Z/(p ^e), then by Lemma 8 the minimal polynomial of $\underline {a}^{\left (s\right )}$ over Z/(p ^e) is unique. Moreover, it can be seen from the process of the proof above that n=k and $g(x)=\prod \limits _{i=0}^{n-1}\left (x-\sigma ^{i} \left (\alpha ^{s}\right ) \right )$, and so g(x) is obviously only depending on f(x). Finally, by Lemma 9 we have

$$per(g(x),p^{e})=ord(\alpha^{s})=\frac{ord(\alpha)}{\gcd \left( ord(\alpha \right),s)}= \frac{per(f(x),p^{e})}{\gcd \left( per(f(x),p^{e}),s\right)}\text{,} $$

thus $per(g(x),p^{e})=\frac {p^{e-1}(p^{n}-1)}{\gcd \left (p^{e-1}(p^{n}-1),s\right )}$ . This completes the proof. □