What is meant by a primitive root in mathematics? I tried Weisstein but I can’t understand it, and Dr. Math only answers high school kid questions. I have no reason for knowing what a number’s primitive root is, except curiosity.
I never heard of this before, but I looked it up and I think I get it.
I will use ** to indicate raising to a power. So 3**2=9. And I’ll use sqrt(x) to indicate the square root of x so sqrt(9)=3.
Consider x**n=1. That is some number x raised to some integer power n equals 1. This makes x a root of unity. The first integer n where this works out is where x is a primitive root of unity.
Consider x=1 and n=1, well 1**1=1 so 1 is a primitive first root of unity.
Now consider x=-1 and n=2, well -1**2=1 and this is the first n where -1 is working out, so -1 is a primitive second root of unity.
The complex number -.5 + sqrt(3)/2i will be a primitive third root of unity. Note that 1 is also a third root of unity, but not a primitive third root.
I only see the term “primitive root” being used in conjunction with unity. But I guess you could extend the concept and talk about primitive roots of other numbers.
I’ve heard it used in talking about roots of unity and multiplicative groups of integers mod n. It’s a number that “generates” all the other numbers (all the other roots of unity or the entire group).
For example, with roots of unity. Say you’re looking at the 12th roots of unity (12 of them). A 12th root of unity x would be one where x, x[sup]2[/sup], x[sup]3[/sup],…,x[sup]12[/sup] gives you all 12th roots of unity. i (square root of -1) is a 12th root of unity, but not a primitive one because the different powers of i only give i,-1,-i, and 1. e[sup]ip/6[/sup] (where p=pi) would be a primitive 12th root of unity.
Or, if you’re familiar with the integers mod n, then a primitive root is a number which generates the multiplicative group of numbers relatively prime to n (and less than n). So, for example in the integers mod 7, 3 is a primitive root, because the powers of 3 are 3,2,6,4,5,1 (all of the integers mod 7 relatively prime to 7). 4 is not a primitive root, since the powers of 4 are 4,2,1–you don’t get all of them. And, for another example, in the integers mod 8, there is no primitive root.
I think here’s a better explanation 
Eric Wesstein’s World of Mathematics: Primitive Root
This site is a good one to keep on your bookmarks by the way. As far as I know, it isn’t lying when it claims to be the most extensive mathematics resource on the web. There are seperate entries for primitive roots of unity too…
And yes, I had no idea what a primitive root was before I looked it up. Frankly speaking, I don’t think I want to know what it is now that I’ve looked it up 