Let’s return to the test for
\(F_n\)’s primality in
Fact 17.5.1. A careful look at the proof shows that
\(3\) is a primitive root for
\(F_n\text{,}\) if
\(F_n\) is prime. Thus, if we had infinitely many Fermat primes (and not just five of them), we’d have an integer which is a primitive root of infinitely many primes.
There is some historical connection as well. Gauss spent some time investigating the patterns of repetitions in simple decimal expansions of fractions, like
\(\frac{1}{3}=.333\ldots \) or
\(\frac{2}{7}=.285714285714\ldots \text{.}\) It turns out that this is directly connected to whether
\(10\) is a primitive root for a given prime (see
Exercise 17.7.21). Likewise, when Euler found that
\(F_5=4294967297\) was composite (recall
Subsection 12.1.1) he would have been helped along quite a bit by information about this conjecture, as his proof looked directly at factors of powers of 2 (plus one) and their possible form, not powers of 3.
We can use these ideas to find another possible way to attack Artin’s Conjecture. It’s not directly related to reciprocity per se, but still connects all our theoretical ideas of the last several sections.
Example 17.5.4.
We put this in the form of several steps. Verifying several facts in these steps is left to
Exercise Group 17.7.8–11.
Recall from the very end of
Section 11.6 that if
\(q\) and
\(p=2q+1\) are both odd primes, then we call
\(q\) a Germain prime. In that case, every residue of
\(p\) other than
\(a=-1\) and
\(a=0\) is a primitive root or a QR. One way to interpret this is as complementing
Fact 16.4.5, which characterizes even powers of a primitive root as being QRs; namely, for
\(p\) nearly all odd powers must be primitive roots.
Such a prime \(p\) must be of the form \(p\equiv 3\text{ (mod }4)\text{.}\) This follows because \(q\) is odd so \(q=2k+1\) for some integer \(k\text{,}\) yielding
\begin{equation*}
p=2(2k+1)+1=4k+3\text{.}
\end{equation*}
(This is how we know that
\(-1\text{,}\) which is clearly not a primitive root, also isn’t a QR; recall
Fact 16.1.2.)
In this case, not only are all residues other than \(0,-1\) either a primitive root or a QR, but \(a\) is one of these things precisely when \(p-a\) is the other. We know that
\begin{equation*}
1^2,2^2,3^2,\ldots,q^2
\end{equation*}
are all different modulo \(p\text{,}\) and of course all of these are QRs (and so not primitive roots).
Here is the key; that means that the additive inverses of perfect squares,
\(p-k^2\text{,}\) for
\(2\leq k\leq q\text{,}\) must all be primitive roots. The smallest of these,
\(p-4\text{,}\) must thus be a primitive root for any such (safe; recall
Subsection 11.6.4) prime
\(p=2q+1\text{.}\)
So if there were infinitely many such Germain primes, we would also have an explicit example of Artin’s conjecture … but, so far, no such luck.
The
largest currently known (as of this writing, discovered in early 2016) Germain prime, due to James Scott Brown, is
\begin{equation*}
2618163402417\cdot 2^{1290000} - 1
\end{equation*}
which is a number with close to four hundred thousand digits. (The previous record had about half as many, so this is a huge advance.)