# Tag Archives: roots

## From primitive roots to Euclid’s orchard

Commenter Snowball pointed out the similarity between Euclid’s Orchard… …and this picture of primitive roots I made a year ago: At first I didn’t see the connection, but Snowball was absolutely right. Once I understood it, I made this little … Continue reading

Posted in pattern, pictures, posts without words | Tagged , , , , , | Comments Off on From primitive roots to Euclid’s orchard

## The Möbius function proof, part 2 (the subset parity lemma)

Continuing from my previous post, we are in the middle of proving that satisfies the same equation as , that is, and that therefore for all , that is, is the sum of all the th primitive roots of unity. … Continue reading

Posted in arithmetic, combinatorics, complex numbers, primes, proof | Tagged , , , , , , , , , | 3 Comments

## The Möbius function proof, part 1

In my last post, I introduced the Möbius function , which is defined in terms of the prime factorization of : if has any repeated prime factors, that is, if is divisible by a perfect square. Otherwise, if has distinct … Continue reading

Posted in Uncategorized | Tagged , , , , , , , , , | 9 Comments

## The Möbius function

Time to pull back the curtain a bit! My recent series of posts on complex roots of unity may seem somewhat random and unmotivated so far, but the fact is that I definitely have a destination in mind—we are slowly … Continue reading

Posted in Uncategorized | Tagged , , , , , , , , | 5 Comments

## Computing sums of primitive roots

Remember this picture? It, and other pictures like it, express the fact that for a given , if we take the primitive roots for each of the divisors of , together they make up exactly the set of all th … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 9 Comments

## Sums of primitive roots

In my previous post, we saw that adding up all the complex th roots of unity always yields zero (unless , in which case the sum is ). Intuitively, this is because the roots are symmetrically distributed around the unit … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 4 Comments

## Sums and symmetry

Let’s continue our exploration of roots of unity. Recall that for any positive integer , there are complex numbers, evenly spaced around the unit circle, whose th power is equal to . These are called the th roots of unity. … Continue reading

Posted in geometry, pictures | Tagged , , , , , , | 2 Comments

## Primitive roots of unity

So we have now seen that there are always different complex th roots of unity, that is, complex numbers whose th power is equal to , equally spaced around the circumference of the unit circle. Consider the first th root … Continue reading

Posted in geometry, pictures | Tagged , , , , , | 4 Comments

## Complex multiplication and roots of unity

If played around with the question from my previous post, you probably found something like the following: That is, as the powers of we get , , and with all possible sign combinations. Of course, since , if we continue … Continue reading

Posted in geometry, pictures | Tagged , , , , , | 4 Comments

## Totient sums

I took a bit of a break to travel to Japan for a conference, but I’m back now to continue the series I started with Post Without Words #10, a follow-up post, and Post Without Words #11. Recall that we … Continue reading

Posted in geometry, pattern, pictures, posts without words, proof | Tagged , , , , , | 4 Comments