Number Theory Help: Homework Equation on Prime Generator

[Kindayr]

Homework Statement

Let $p$ be an odd prime. Show that there exists $a\in\mathbb{Z}$ such that $[a]\in\mathbb{Z}^{\times}_{p}$ is a generator and $$a^{p-1}=1+cp$$ for some $c$ coprime to $p$.

Homework Equations

The Attempt at a Solution

I honestly have no idea where to even start with this. Any help will be much appreciated, no full solutions though!

 

[Dick]

You could start with Fermat's Little Theorem.

 

[Kindayr]

You could start with Fermat's Little Theorem.

Well I know its obvious that $a^{p-1}\equiv 1 (\mod p)$ since $[a]\in\mathbb{Z}_{p}^{\times}$. Hence there exists $k\in\mathbb{Z}$ such that $a^{p-1}=1+kp$. How do I show that $\gcd (k,p)=1$?

 

[Dick]

Well I know its obvious that $a^{p-1}\equiv 1 (\mod p)$ since $[a]\in\mathbb{Z}_{p}^{\times}$. Hence there exists $k\in\mathbb{Z}$ such that $a^{p-1}=1+kp$. How do I show that $\gcd (k,p)=1$?

Offhand, I don't know. You were asking for a hint to start and no full solutions. I'll think about it though, if you don't find the solution first.

 

[Deveno]

for [a] to be a generator, we also need to know that p-1 is the smallest positive integer k for which:

a^k ≡ 1 (mod p)

i suggest asking yourself a similar, but related question:

suppose d|(p-1). how many elements of Z_p^x are of order d?

hint: suppose a is of order d.

(i suppose i "should write" [a] instead of a. i mean the modulo class, NOT the integer)

look at the set {1,a,a²,...,a^d-1}.

this is a cyclic group, how many generators does it have?

hint #2: eventually, you'll have to drag in the totient function into this. why did i suggest looking at those d that divide p-1?

 

[Kindayr]

hint: suppose a is of order d.

(i suppose i "should write" [a] instead of a. i mean the modulo class, NOT the integer)

look at the set {1,a,a²,...,a^d-1}.

this is a cyclic group, how many generators does it have?

Any element $a^{r}$ such that $\gcd(r,d)=1$ is a generator.

 

[Deveno]

Any element $a^{r}$ such that $\gcd(r,d)=1$ is a generator.

yes, that is true. and how many of those "r's" are there?

 

[Kindayr]

hint #2: eventually, you'll have to drag in the totient function into this. why did i suggest looking at those d that divide p-1?

The size of $\mathbb{Z}_{p}^{\times}$ is $\phi (p)=p-1$

 

[Kindayr]

yes, that is true. and how many of those "r's" are there?

$\phi (d)$ many.

 

[Deveno]

The size of $\mathbb{Z}_{p}^{\times}$ is $\phi (p)=p-1$

yes, that is also true, and you'll want to use that fact later on. how does "d" enter into this?

$\phi (d)$ many.

exactly! now...given any a of order d, how many "other elements" of order d, do we get "for free" along with it?

and...do you know of any way to relate φ(d) and φ(p-1)?

 

[Kindayr]

yes, that is also true, and you'll want to use that fact later on. how does "d" enter into this?

If we take $d=p-1$, then there are only $\phi (p-1)$ many generators of $\mathbb{Z}_{p}^{\times}$. Is this right?

do you know of any way to relate φ(d) and φ(p-1)

Since $d|p-1$, it follows that $\phi(d)|\phi(p-1)$. Hence there exists $k\in\mathbb{Z}$ such that $\phi(p-1)=k\phi(d)$.

 

[Deveno]

If we take $d=p-1$, then there are only $\phi (p-1)$ many generators of $\mathbb{Z}_{p}^{\times}$. Is this right?

we only know that if we already know that there is in fact an element a of order p-1, which is what we're trying to prove!

we don't even know if there are any elements of order d, but if there are, how many must we have (at least)?

ok, let's step back for a minute, and see if you can glimpse where we're going:

every element of Z_p^x has SOME order, and we know that the order of any element has to divide p-1, by a theorem of Lagrange.

let's call the number of elements of order d, f(d).

so if the divisor list of p-1 is 1 = d₁,d₂,...,d_k = p-1

p -1 = f(d₁) + f(d₂) +...+ f(d_k)

$$= \sum_{d|p-1} f(d)$$

for every d that divides p-1, f(d) is either ___ or _____ ?

 

[Kindayr]

for every d that divides n, f(d) is either ___ or _____ ?

For every $d|p-1$, it follows that every $f(d)$ is either $\phi(d)$ or $0$, because we haven't verified the fact that there exists an element of order $d$ for each $d|p-1$.

 

[Deveno]

For every $d|p-1$, it follows that every $f(d)$ is either $\phi(d)$ or $0$, because we haven't verified the fact that there exists an element of order $d$ for each $d|p-1$.

great! glad to see you're still with me on this.

now, do you know the answer to this question?

$$\sum_{d|p-1} \varphi(d) = ?$$

 

[Kindayr]

great! glad to see you're still with me on this.

now, do you know the answer to this question?

$$\sum_{d|p-1} \varphi(d) = ?$$

Yep :)
$$\sum_{d|p-1}\phi(d)=p-1$$

 

[Deveno]

and is it not true that for every d that divides p-1:

f(d) ≤ φ(d)?

 

[Kindayr]

and is it not true that for every d that divides p-1:

f(d) ≤ φ(d)?

Yep since $f(d)$ is either $\phi(d)$ or $0$; both of which are less than or equal to $\phi(d)$. And since $\sum_{d|p-1}f(d)=\sum_{d|p-1}\phi(d)$, it follows from the previous sentence that $f(d)=\phi(d)$. So now we know that $\phi(d)$ is exactly the number of elements in $\mathbb{Z}_{p}^{\times}$ that have order $d$. Hence, for each $d|p-1$, there exists exactly $\phi(d)$ many (which is non-zero) elements in $\mathbb{Z}_{p}^{\times}$ whose order is $d$.

 

[Deveno]

Yep since $f(d)$ is either $\phi(d)$ or $0$; both of which are less than or equal to $\phi(d)$. And since $\sum_{d|p-1}f(d)=\sum_{d|p-1}\phi(d)$, it follows from the previous sentence that $f(d)=\phi(d)$. So now we know that $\phi(d)$ is exactly the number of elements in $\mathbb{Z}_{p}^{\times}$ that have order $d$. Hence, for each $d|p-1$, there exists exactly $\phi(d)$ many (which is non-zero) elements in $\mathbb{Z}_{p}^{\times}$ whose order is $d$.

so now we've established existence for EVERY d that divides p-1. and one of those divisors is...?

 

[Kindayr]

so now we've established existence for EVERY d that divides p-1. and one of those divisors is...?

We can take $d=p-1$ and hence there exists $\phi(p-1)$ many elements of $\mathbb{Z}_{p}^{\times}$ whose order is $p-1$. But since the order of $\mathbb{Z}_{p}^{\times}$ is $p-1$, it follows that these elements are exactly the generators of $\mathbb{Z}_{p}^{\times}$. Hence, we've show that there exists at least a single generator of $\mathbb{Z}_{p}^{\times}$.

So let $a\in\mathbb{Z}_{p}^{\times}$ be one of these generators. By Fermat's Little Theorem we have $a^{p-1}\equiv 1$ $\mod {p}$. Hence there exists $k\in\mathbb{Z}$ such that $a^{p-1}=1+kp$.

 

[Deveno]

We can take $d=p-1$ and hence there exists $\phi(p-1)$ many elements of $\mathbb{Z}_{p}^{\times}$ whose order is $p-1$. But since the order of $\mathbb{Z}_{p}^{\times}$ is $p-1$, it follows that these elements are exactly the generators of $\mathbb{Z}_{p}^{\times}$. Hence, we've show that there exists at least a single generator of $\mathbb{Z}_{p}^{\times}$.

indeed, we've shown that there's usually a lot of generators.

there's just one more matter to clear up.

we know that there is some a with order p-1, so of course a^p-1 = 1 + kp.

but...we haven't yet shown gcd(k,p) = 1. any thoughts?

 

[Kindayr]

but...we haven't yet shown gcd(k,p) = 1. any thoughts?

Well if $1\le k<p$, then clearly $\gcd(k,n)=1$. But if $p<k$, then its not so clear.

 

[Deveno]

Well if $1\le k<p$, then clearly $\gcd(k,n)=1$. But if $p<k$, then its not so clear.

true enough. but we haven't used all of the help we were given by the problem at the start.

p is an ODD prime. maybe that is important somehow...hmm, if p is odd, then p-1 is even, so:

a^p-1 = 1 + kp

but p-1 = 2m, so:

(a^m)² - 1 = kp

can you do anything with that?

 

[Kindayr]

true enough. but we haven't used all of the help we were given by the problem at the start.

p is an ODD prime. maybe that is important somehow...hmm, if p is odd, then p-1 is even, so:

a^p-1 = 1 + kp

but p-1 = 2m, so:

(a^m)² - 1 = kp

can you do anything with that?

Well we can say $kp=(a^{m})^2-1=(a^{m}-1)(a^{m}+1)$, or is there something that I'm missing?

 

[Deveno]

Well we can say $kp=(a^{m})^2-1=(a^{m}-1)(a^{m}+1)$, or is there something that I'm missing?

that's a good start. now what can we say about divisibility of the factors on the right by p, because p is prime?

 

[Kindayr]

that's a good start. now what can we say about divisibility of the factors on the right by p, because p is prime?

Either $p|(a^{m}+1)$ or $p|(a^{m}-1)$. If it is the latter, then $p|(a^{m}-1)$ implies that $a^{m}\equiv 1\mod p$, but that would mean that the order of $a$ is $m$, which contradicts the fact that $a$ is a generator of $\mathbb{Z}_{p}^{\times}$. Hence, it must be that $a^{m}\equiv -1\mod p$.

 

[Deveno]

Either $p|(a^{m}+1)$ or $p|(a^{m}-1)$. If it is the latter, then $p|(a^{m}-1)$ implies that $a^{m}\equiv 1\mod p$, but that would mean that the order of $a$ is $m$, which contradicts the fact that $a$ is a generator of $\mathbb{Z}_{p}^{\times}$. Hence, it must be that $a^{m}\equiv -1\mod p$.

well, you know, that's ok, but the thing is, we don't really care which factor p divides, so long as we know that p divides one of them. the point is:

k = (a^m - 1)(a^m + 1)/p

we're trying to prove something about k, here: gcd(k,p) = 1.

could k be 0? well, that would mean one of our factors is 0, but surely a > 1.

and if 1 ≤ k < p, as you yourself already noted, gcd(k,p) = 1.

so if k ≥ p, the only way gcd(k,p) ≠ 1, is if p|k.

and that's what we want to show "can't happen".

now, for ANY positive integer n, and any ODD prime, can p divide both n and n+2?

 

[Kindayr]

No, for any odd prime $p$ it is not the case for both $p|n$ and $p|n+2$, for any $n\in\mathbb{Z}$.

 

[Kindayr]

I'm still stuck on your last hint :(

 

[Kindayr]

OOOOOH

Since $p$ odd prime it follows that it cannot be that both $p|a^{m}-1$ and $p|a^{m}+1$. I understand why we want to do this, but what if $p^2|a^{m}+1$?

 

[Deveno]

last night, i started thinking about this some more, and i am wondering whether it is even true (that k is co-prime to p).

in particular i turned up this:

5 is a primitive root (generator) mod 40487 but:

5⁴⁰⁴⁸⁶ ≡ 1 (mod 40487²)

(http://mathoverflow.net/questions/2...tive-root-modulo-p-a-primitive-root-modulo-p2)

is that exactly what the problem you were given says?