 In 1796, Carl Friedrich Gauss discovered the regular 17-gon, the heptidecagon, could be constructed using compass and straight-edge techniques only. It was the first new compass and straight-edge construction in 2,000 years, and because of it Gauss decided to study mathematics instead of linguistics. Gauss summarized his work in the last section of Disquisiciones Aritmeticae 1801. So Gauss knew that constructing a regular n-gon relies on finding the values of the cosine of 2 pi over n and sine of 2 pi over n. And Gauss knew these could be obtained from the complex solutions to the equation z to power n minus 1. And that comes from De Mois' formula because the solutions to z to power n minus 1 will be the complex numbers of the form cosine 2 pi over nk plus i sine 2 pi over nk for k varying between 0 up to n minus 1. Now one of these roots is real, namely if k equals 0 we get z equals 1. And since z equals 1 is a root, then z minus 1 is a factor and we can remove it. And because this equation is useful for constructing the regular n-gon, it's known as the cyclatomic equation from the Greek words meaning circle cutting. And we also say that the solutions are the roots of unity. Now earlier in the Disquisiciones Gauss defined what a primitive root was, however his definition is slightly different from the one we now use and for that matter he used in this context and so we'll define it the way that it was used. R is a primitive nth root of unity if n is the least power of r equal to 1. For example let's find the fourth roots of unity and determine which ones are primitive. So the fourth roots of unity are the solutions to z to the fourth minus 1. And we can solve these by factoring z to the fourth minus 1 factors and so either z squared minus 1 is equal to 0 or z squared plus 1 is equal to 0. Solving these gives us and so we have four solutions 1, negative 1, i, negative i. Now while these are all fourth roots, in order to be a primitive fourth root the least power that makes 1 must be the fourth power. Since it's minus 1 squared is equal to 1 and 1 to the 1th is equal to 1 neither is primitive. On the other hand, since the least power of i or negative i that gives 1 is the fourth power then both of these are primitive. So why do we care? Well there's two important features about primitive roots. If r is a primitive root then r squared r cubed and so on the other powers of r will give the other roots and the sum of all the powers of r plus 1 is going to be equal to 0. Both of these are easy to prove. Now while the powers of a primitive root will generate all roots in general the powers of a primitive root will not always be primitive roots. For example i is a primitive fourth root but i squared equals negative 1 is not primitive. But if r to the k is primitive then r to the k to the n will also be primitive for all n. Now these superscripts become unreadable very quickly so Gauss introduced the notation. r to the power k is bracket k. So we know a couple of things. Bracket 0 is equal to 1 and bracket n is also equal to 1. If i multiply bracket p by bracket q i get bracket p plus q and if i raise bracket p to the q-th power i get bracket pq. All of these are really restatements of the standard rules of exponents where bracket k is r to the power k. You should prove these. So Gauss considered the sequence beginning bracket 1, bracket g, bracket g squared and so on where the indices are powers of g. There are two possibilities. First, the sequence might include all of the roots. Second, the sequence might only include some of the roots. For example let r be a primitive fifth root of unity. Let's find the distinct terms of the sequence beginning bracket 1, bracket 2 and the terms of the sequence beginning bracket 1, bracket 4. So the sequence beginning bracket 1, 2 will have terms whose indices are powers of 2. Well we can just write those down. Now while the sequence has an infinite number of terms, some of these can be reduced. Since bracket k is r to the power k where r is a primitive fifth root of unity, we note the following. Bracket 8, well that's really r to the 8th and since r is a primitive fifth root of unity then I know what r to the 5th is. So we'll split that into r to the 5th times r to the 3rd. And remember we did this because we knew that r is a primitive fifth root of unity. So r to the 5th is just 1 and we can omit that factor. And r to the 3rd is the same as bracket 3. Now it should be clear that we can't reduce bracket 1, 2 or 4 this way so those three terms remain unchanged. And so this term bracket 8 is really bracket 3. Similarly our next term bracket 16, well that's really r to the 16th splitting off factors of r to the 5th simplifying and rewriting. r to the 32nd we can rewrite that as and bracket 64, well that's r to the 64th which is a whole bunch of r to the 5th times r to the 4th. Again since we're dealing with primitive fifth roots of unity all of those r to the 5th can simplify down to 1 and so we get bracket 4. And so our sequence reduces to brackets 1, 2, 4, 3, 1, 2, 4 and so on. And we'll note that this contains all of the roots. If we take a look at the sequence beginning with 1, 4 we'll have terms whose indices are powers of 4 and we can reduce those. Bracket 16 becomes bracket 1 and we've already found bracket 64. And so this sequence is going to reduce.