Roots on a Circle

Number Theory Level 5

Denote by $N$ the number of pairs of positive integers $(n,m),$ such that $m<n\leq 100$ and the polynomial $x^n+x^m+1$ has a root on the unit circle. Find the last three digits of $N.$

Details and assumptions

A root on the unit circle is a complex root at the distance exactly $1$ from the origin.

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.