MathLabs

Problem 6

Prove that there exists a convex 1990-gon whose angles are all equal and whose side lengths are 12,22,…,199021^2,2^2,\ldots,1990^2 in some order.
Step 2 of 4: Factor the root of unity into orders 2, 5, and 199
Previous stepNext step
1990=2⋅5⋅199,ω=−ab1990=2\cdot5\cdot199,\qquad \omega=-ab
Detailed analysis

Choose a primitive fifth root a and a primitive 199th root b so that omega equals minus a b. Every 1990th-root direction is uniquely represented as (-1)^i a^j b^k with 0≤i<2, 0≤j<5, and 0≤k<199.