The Complex Tale of Eisenstein Prime Numbers

By

Home / The Complex Tale of Eisenstein Prime Numbers

Visual representation of Eisenstein Primes: Image by Johannes Rössel

Last week’s “Several Different Paths to Prime Numbers” opened with this intriguing image.

Unfortunately, there was no room to answer the question “What are Eisenstein primes?”

An Explanation of an Eisenstein Prime Number

We already know that a “prime number” is a number that can only be evenly divided by itself and the number one.

An “Eisenstein Integer” is a complex number constructed with one of the cubic roots of one. An Eisenstein integer have the form “a + b*ω”, where ‘a’ and ‘b’ are integers, and “ω (the lower-case Greek omega) = (1/2)*( i * ( 3^(1/2) ) – 1 )”. In this equation, ‘i’ is the imaginary square root of negative one.

Review of Imaginary Numbers and Complex Numbers

As noted in “The Definitive Quick Reference Guide to All Types of Numbers“, the imaginary number ‘i’ is defined as “i = (-1)^(1/2)”. Thus, “i^2 = i*i = -1″.

The set of Imaginary numbers is the set {y*i} where ‘y’ is a Real number.

Complex numbers take the form “x + y*i} where both ‘x’ and ‘y’ are Real numbers. One might prefer to exclude the value “y = zero”, because this leaves a simple Real number. However, Eisenstein integers “a + b*ω” includes cases where b=zero, because they are very useful…as we will see in one of the “Eisenstein Prime” families.

A Second Look at the Cube Roots of One

Within the Real numbers, (1)^(1/3) = 1. Eisenstein decided to examine the complex roots.

Multiplying ‘ω’ by itself gives ω^2 =

  • = ((1/2)*( i * ( 3^(1/2) ) – 1 )) * ((1/2)*( i * ( 3^(1/2) ) – 1 )) =
  • = (1/4)*( i*i*( 3^(1/2) )*( 3^(1/2) ) – 2*i*( 3^(1/2) ) + 1 ) =
  • = (1/4)*( -1*3 – 2*i*( 3^(1/2) ) + 1 ) =
  • = (1/4)*( 1 – 3 – 2*( i*( 3^(1/2) ) ) ) =
  • = (1/4)*(-2  – 2*( i*( 3^(1/2) ) ) ) =
  • = (-1/2)*( i * ( 3^(1/2) ) + 1)

Then ω^3 =

  • = ω*ω^2 =
  • = (1/2)*( i * ( 3^(1/2) ) – 1 ) * (-1/2)*( i * ( 3^(1/2) ) + 1) =
  • = (-1/4)*( (i*i* ( 3^(1/2) )*( 3^(1/2) + i * ( 3^(1/2) ) – i * ( 3^(1/2) ) – 1 ) =
  • = (-1/4)*( -3 – 1) =
  • = (-1/4)*(-4) =
  • = +1

(Those equations are much easier to write using pen and paper than in Notepad.) So ω^3 = 1 in keeping with the earlier claim that ω = 1^(1/3).

Are Eisenstein Integers Important?

Eisenstein integers are indeed important as regards mathematical constructs known as fields and rings. The set of Eisenstein integers is closed under addition and multiplication.

However, this article must return to the topic of Eisenstein primes.

What is an Eisenstein Prime Number?

Eisenstein Prime Numbers and Units – Image by Januszkaja

After the discussion of complex numbers, it is worth the reminder that an Eisenstein integer, ‘e’, has the form “e = a + b*ω” where ‘a’ and ‘b’ are integers. So although an Eisenstein “Integer” does have an imaginary component, ‘ω’, with an irrational square root of three for good measure, the ‘a’ and ‘b’ portions are integers.

With that in mind, the definition of an Eisenstein Prime Number is very similar to that of primes in the set of Natural numbers.

Finally, the Definition of an Eisenstein Prime Numbers

An Eisenstein Prime cannot be expressed as the product of other Eisenstein Integers. The only factors are the Eisenstein integer itself (“a + bω”), its conjugate (“a – bω”), and the Eisenstein “unit” numbers 1, ω, ω^2, and their negatives.

Three Types of Eisenstein Prime Numbers

One type has a single example. The other types are families of primes.

The Loneliest Eisenstein Prime: The simplest Eisenstein Prime is “1 – ω”.

Eisenstein Primes in the Natural NumbersOne family of Eisenstein Primes is simply “all ‘a’ where ‘a’ is a prime number and 3 divides (a – 2) evenly”. The value of ‘b’ is zero in the expression “a + b*ω”. Therefore this family of Eisenstein Prime Numbers also appears to be a subset of the Natural numbers: there is no complex ‘ω’ component.

  • These primes begin as {2, 5, 11, 17…}.

The Most Complex Eisenstein PrimesThe other family of primes takes the form “a + b*ω” or “a + b*ω*ω”, with two criteria for the product “(a + b*ω)*(a + b*ω*ω)”.

  • (a + b*ω)*(a + b*ω*ω) =
  • = a^2 + (a*b*ω + a*b*ω*ω) + (b^2)*(ω^3) =
  • = a^2 + (a*b)*(ω + ω^2) + b^2″ .

Note that “ω + ω^2 = (1/2)*( i * ( 3^(1/2) ) – 1 ) + (-1/2)*( i * ( 3^(1/2) ) + 1) = -1.

Therefore the previous paragraph’s equation simplifies nicely.

  • a^2 + (a*b)*(ω + ω^2) + b^2 =
  • = a^2 + (a*b)*(-1) + b^2 =
  • = a^2 – a*b + b^2″.

The second criterion is that, when calculating the integer “n = (a + b*ω)*(a + b*ω*ω) = a^2 – a*b + b^2″, then ‘n’ must be a prime number and 3 must divide (n – 1) evenly.

A Grid of Eisenstein Primes

"Eisentstein Primes on Grid" by Fropuff

“Eisentstein Primes on Grid” by Fropuff

This image shows the two families of Eisenstein primes. Points on the green axes show one family; the other family lives between the axes.

The Importance of Eisenstein Primes

Perhaps the most practical use for Eisenstein primes is the ongoing search for very large prime numbers. The Eisenstein Prime rules help discover large primes other than the Mersenne primes, for example.

Other than that, Eisenstein primes appear in heavy-duty mathematics that cannot be summarized at the end of a lengthy article. Here is an example from a “Pure Mathematics” seminar description:

“Cecilia Busuiok: Einstein congruences and K-theory:
In this talk, we will investigate congruences of periods of parabolic modular forms at Eisenstein primes and their connection to the arithmetic of the K2-groups of rings of integers. Our approach to this study deals with an explicit construction of Eisenstein cohomology classes in the parabolic cohomology of the arithmetic group ¡0(N).”

References:

Department of Mathematics, Royal Holloway, University of London. “Pure mathematics seminar spring 2011.” (2011). Accessed Oct. 31, 2011.

Weisstein, Eric W. MathWorld-A Wolfram Web Resource. “Eisenstein Integer,” “Eisenstein Prime.” (1999-2011). Accessed Oct. 30, 2011.

Leave a Comment