2008年(47)
分类:
2008-11-24 13:46:57
All square roots are periodic when written as continued fractions and can be written in the form:
![]() |
1 ![]() |
||
a1 + | 1 ![]() |
||
a2 + | 1 ![]() |
||
a3 + ... |
For example, let us consider 23:
![]() ![]() |
1 ![]() |
= 4 + | 1 ![]() |
|
1 ![]() ![]() |
1 + | ![]() ![]() 7 |
If we continue we would get the following expansion:
![]() |
1 ![]() |
|||
1 + | 1 ![]() |
|||
3 + | 1 ![]() |
|||
1 + | 1 ![]() |
|||
8 + ... |
The process can be summarised as follows:
a0 = 4, | 1 ![]() ![]() |
= | ![]() ![]() 7 |
= 1 + | ![]() ![]() 7 |
|
a1 = 1, | 7 ![]() ![]() |
= | 7( ![]() ![]() 14 |
= 3 + | ![]() ![]() 2 |
|
a2 = 3, | 2 ![]() ![]() |
= | 2( ![]() ![]() 14 |
= 1 + | ![]() ![]() 7 |
|
a3 = 1, | 7 ![]() ![]() |
= | 7( ![]() ![]() 7 |
= 8 + | ![]() |
|
a4 = 8, | 1 ![]() ![]() |
= | ![]() ![]() 7 |
= 1 + | ![]() ![]() 7 |
|
a5 = 1, | 7 ![]() ![]() |
= | 7( ![]() ![]() 14 |
= 3 + | ![]() ![]() 2 |
|
a6 = 3, | 2 ![]() ![]() |
= | 2( ![]() ![]() 14 |
= 1 + | ![]() ![]() 7 |
|
a7 = 1, | 7 ![]() ![]() |
= | 7( ![]() ![]() 7 |
= 8 + | ![]() |
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?