Visualizing Prime Numbers on the Euler Circle

To gain deeper in­tu­ition into the nature of prime num­bers, we can use the Euler circle, also known as the com­plex unit circle in the com­plex plane.

The unit circle con­sists of all com­plex num­bers $z$ of the form:

where $\theta$ is a real number be­tween $0$ and $2\pi$, $i$ is the imag­i­nary unit ($i^2 = -1$), and $\exp$ de­notes the ex­po­nen­tial func­tion. The unit circle has radius $1$ and is cen­tered at the origin.

Roots of Unity and Primes

An im­por­tant con­cept in both group theory and number theory is the $n$th roots of unity. These are the com­plex so­lu­tions to the equa­tion:

All $n$th roots of unity are equal­ly spaced points on the unit circle, given by:

where $k$ is an in­te­ger with $0 \leq k < n$.

For ex­am­ple, if $n = 5$, the five 5th roots of unity are:

These points are ver­tices of a reg­u­lar pen­ta­gon in­scribed in the unit circle.

Primes and the Unit Circle

If $p$ is a prime number, the $p$th roots of unity divide the circle into $p$ equal­ly spaced, ir­re­ducible points. These points cor­re­spond to the el­e­ments of the cyclic group $\mathbb{Z}_p$ under ad­di­tion modulo $p$.

In this con­text, each gen­er­a­tor $k$ of the group $\mathbb{Z}_p$ cor­re­sponds to a ro­ta­tion by an angle of $2\pi k/p$. Since $p$ is prime, every nonze­ro $k$ is co­prime to $p$ and acts as a gen­er­a­tor, so it­er­at­ing this ro­ta­tion will visit all nonze­ro points on the circle before re­turn­ing to the start­ing point.

Seeing Primes on the Euler Circle

• To vi­su­al­ize a prime $p$ on the Euler circle, plot the $p$th roots of unity:

• Each $z_k$ rep­re­sents a unique el­e­ment of the group $\mathbb{Z}_p$.
• These points cannot be con­struct­ed by com­bin­ing small­er sub­di­vi­sions of the circle (that is, the $p$th roots of unity are not also $n$th roots of unity for any $n < p$), re­flect­ing the pri­mal­i­ty of $p$.

Ex­am­ple: Prime $p = 7$

Plot­ting

yields seven points, each spaced by an angle of $2\pi/7$ ra­di­ans, form­ing a reg­u­lar hep­ta­gon on the unit circle. Each point is ir­re­ducible in the sense that it cannot be reached by re­peat­ing a small­er step that di­vides $2\pi$ evenly (other than the triv­ial one).

In­di­vis­i­ble Points

Prime num­bers, when vi­su­al­ized on the Euler circle as roots of unity, appear as sets of in­di­vis­i­ble, equal­ly spaced points. This geo­met­ric per­spec­tive re­flects their role as the fun­da­men­tal "build­ing blocks" of number theory: they create the sim­plest, most sym­met­ri­cal di­vi­sions of the circle, and thus of the math­e­mat­i­cal uni­verse.