All square roots are periodic when written as continued fractions and can be written in the form:
N = a_{0} + 
1 

a_{1} + 
1 

a_{2} + 
1 

a_{3} + ... 
For example, let us consider 23:
23 = 4 + 23 — 4 = 4 + 
1 
= 4 + 
1 

1 23—4 
1 + 
23 – 3 7 
If we continue we would get the following expansion:
23 = 4 + 
1 

1 + 
1 

3 + 
1 

1 + 
1 

8 + ... 
The process can be summarised as follows:
a_{0} = 4, 
1 23—4 
= 
23+4 7 
= 1 + 
23—3 7 

a_{1} = 1, 
7 23—3 
= 
7(23+3) 14 
= 3 + 
23—3 2 

a_{2} = 3, 
2 23—3 
= 
2(23+3) 14 
= 1 + 
23—4 7 

a_{3} = 1, 
7 23—4 
= 
7(23+4) 7 
= 8 +  23—4  
a_{4} = 8, 
1 23—4 
= 
23+4 7 
= 1 + 
23—3 7 

a_{5} = 1, 
7 23—3 
= 
7(23+3) 14 
= 3 + 
23—3 2 

a_{6} = 3, 
2 23—3 
= 
2(23+3) 14 
= 1 + 
23—4 7 

a_{7} = 1, 
7 23—4 
= 
7(23+4) 7 
= 8 +  23—4 
It can be seen that the sequence is repeating. For conciseness, we use the notation 23 = [4;(1,3,1,8)], to indicate that the block (1,3,1,8) repeats indefinitely.
The first ten continued fraction representations of (irrational) square roots are:
2=[1;(2)], period=1
3=[1;(1,2)], period=2
5=[2;(4)], period=1
6=[2;(2,4)], period=2
7=[2;(1,1,1,4)], period=4
8=[2;(1,4)], period=2
10=[3;(6)], period=1
11=[3;(3,6)], period=2
12= [3;(2,6)], period=2
13=[3;(1,1,1,1,6)], period=5
Exactly four continued fractions, for N 13, have an odd period.
How many continued fractions for N 10000 have an odd period?
These problems are part of Project Euler and are licensed under CC BYNCSA 2.0 UK