All square roots are periodic when written as continued fractions and can be written in the form:
For example, let us consider ?23:
If we continue we would get the following expansion:
The process can be summarised as follows:
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:
?12= [3;(2,6)], period=2
Exactly four continued fractions, for N <= 13, have an odd period.
How many continued fractions for N <= 10000 have an odd period?
(see full solution here).
Based on the algorithm on continued fractions from Wikipedia, we can implement the expansion algorithm as:
From the Wikipedia page:
The algorithm can also terminate on ai when ai = 2 a0, which is easier to implement.
which corresponds to the termination condition we have in the Repeat active pattern (which also checks if the accumulator is empty):
Also, this algorithm doesn’t work on numbers that are perfect squares, i.e. 4, 9, 16, … hence we need to exclude them when searching for our answer.
Here’s the solution in full:
This solution took 92ms to execute on my machine.
Enjoy what you’re reading? Subscribe to my newsletter and get more content on AWS and serverless technologies delivered straight to your inbox.
I’m an AWS Serverless Hero and the author of Production-Ready Serverless. I have run production workload at scale in AWS for nearly 10 years and I have been an architect or principal engineer with a variety of industries ranging from banking, e-commerce, sports streaming to mobile gaming. I currently work as an independent consultant focused on AWS and serverless.
In this course, we’ll cover everything you need to know to use AWS Step Functions service effectively. Including basic concepts, HTTP and event triggers, activities, design patterns and best practices.
Here is a complete list of all my posts on serverless and AWS Lambda. In the meantime, here are a few of my most popular blog posts.
- Lambda optimization tip – enable HTTP keep-alive
- You are thinking about serverless costs all wrong
- Many faced threats to Serverless security
- We can do better than percentile latencies
- I’m afraid you’re thinking about AWS Lambda cold starts all wrong
- Yubl’s road to Serverless
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- AWS Lambda – compare coldstart time with different languages, memory and code sizes
- Guys, we’re doing pagination wrong