
232 Regular Polygons
If m and n are coprime, then there exist integers a, b such that am + bn = 1.
Therefore
1
mn
= a
1
n
+ b
1
m
Hence from angles 2π/m and 2π/n we can construct 2π/mn, and from this we obtain
a regular mn-gon.
Corollary 20.7. Suppose that n = p
m
1
1
... p
m
r
r
where p
1
,..., p
r
are distinct primes.
Then n is constructive if and only if each p
m
j
j
is constructive.
Another obvious result:
Lemma 20.8. For any positive integer m, the number 2
m
is constructive.
Proof. Any angle can be bisected by ruler and compasses, and the result follows by
induction on m.
This reduces the problem of constructing regular polygons to the case when the
number of sides is an odd ...