Cyclotomic Polynomials Revisited

It was in November of 2020 that I made my first and only post about cyclotomic polynomials. Here is a link to that post. Today I was reminded of this category of polynomials because the number associated with my diurnal age today, 28057, is connected with the third cyclotomic polynomial, namely \(x^2 + x + 1\) because the number is a member of OEIS A053183:

  OEIS A053183: primes of the form \(p^2 + p + 1\) when \(p\) is prime.

In the case of 28057, the value of \(p\) is 167 and thus:$$28057 = 167^2 + 167 + 1$$What follows is what Gemini had to say about this particular polynomial.

The Complex Roots of $x^2 + x + 1 = 0$

The roots of the polynomial $x^2 + x + 1$ are the two primitive cube roots of unity. In mathematical literature, these are commonly denoted by the Greek letters ω (omega) and ω².

1. Numerical Values

Using the quadratic formula, we find the exact complex coordinates:

• x₁: $-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i$ (approx. $-0.5 + 0.866i$)
  
  
• x₂: $-\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i$ (approx. $-0.5 - 0.866i$)

2. Geometric Representation

When plotted on the Argand plane (complex plane), these roots exhibit specific geometric properties:

• The Unit Circle: Both roots have a magnitude of 1, meaning they sit exactly on the circumference of a circle with radius 1 centered at the origin.
  
  
• Conjugate Symmetry: The roots are reflections of each other across the real (horizontal) axis.
  
  
• The Equilateral Triangle: These two roots, combined with the third cube root ($x = 1$), form the vertices of a perfect equilateral triangle.

For further exploration of complex numbers and their visualization, you can use tools like the GeoGebra Complex Number Tool or check detailed proofs on Wolfram MathWorld.

All numbers generated by \(x^2+x+1\) for positive integer values of \(x\) are one more than the pronic numbers because the polynomial can be written as \(x(x+1) + 1 \).