MathDB

\omega

is a complex number such that

\omega^{2013} = 1

and

\omega^m \neq 1

for

m=1,2,\ldots,2012

. Find the number of ordered pairs of integers

(a,b)

with

1 \le a, b \le 2013

such that

\frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3}

is the root of some polynomial with integer coefficients and leading coefficient

1

. (Such complex numbers are called algebraic integers.)
Victor Wang

48