Tag Archives: primitive

From primitive roots to Euclid’s orchard

Posted on July 26, 2017 by Brent

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 algorithm, Euclidean, gcd, orchard, primitive, roots | Comments Off

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

Posted on December 3, 2016 by Brent

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 circle, complex, moebius, mu, primitive, proof, roots, sum, unit, unity | 3 Comments

The Möbius function proof, part 1

Posted on November 29, 2016 by Brent

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 circle, complex, moebius, mu, primitive, proof, roots, sum, unit, unity | 9 Comments

The Möbius function

Posted on November 22, 2016 by Brent

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 circle, complex, moebius, mu, primitive, roots, sum, unit, unity | 5 Comments

Computing sums of primitive roots

Posted on November 15, 2016 by Brent

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 circle, complex, primitive, roots, sum, unit, unity | 9 Comments

Sums of primitive roots

Posted on November 9, 2016 by Brent

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 circle, complex, primitive, roots, sum, unit, unity | 4 Comments

Primitive roots of unity

Posted on October 20, 2016 by Brent

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 circle, complex, primitive, roots, unit, unity | 4 Comments

Totient sums

Posted on October 7, 2016 by Brent

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 Euler, phi, primitive, roots, sum, unity | 4 Comments

Post without words #11

Posted on September 8, 2016 by Brent

Posted in geometry, pattern, pictures, posts without words, proof | Tagged Euler, phi, primitive, roots, sum, unity | 7 Comments

A few words about PWW #10

Posted on August 30, 2016 by Brent

If you still want to think more about the picture in my previous post, stop reading now! Here’s a simple way to think about how the picture is made, as noted by Fergal Daly. The th circle (starting with ) … Continue reading →

Posted in geometry, pattern, pictures, posts without words | Tagged gcd, primitive, roots, unity | 5 Comments