Project Euler – Problem 57 Solution


It is possible to show that the square root of two can be expressed as an infinite continued fraction.


By expanding this for the first four iterations, we get:

1 + 1/2 = 3/2 = 1.5

1 + 1/(2 + 1/2) = 7/5 = 1.4

1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666…

1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379…

The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.

In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?


// define function to return all the numerator-denominator pairs for the first n expand
let expand n =
    Seq.unfold (fun (num, denom) -> Some((num, denom), (denom*2I+num, denom+num))) (3I, 2I)
    |> Seq.take n

let answer =
    expand 1000
    |> Seq.filter (fun (num, denom) -> num.ToString().Length > denom.ToString().Length)
    |> Seq.length

If you look at the patterns 3/2, 7/5, 17/12, 41/29, and so on, it’s easy to spot a pattern where the numerator and denominator of iteration n can be derived from the iteration n-1:

Numerator(n) = Numerator(n-1) + 2 * Denominator(n-1)

Denominator(n) = Numerator(n-1) + Denominator(n-1)

Liked this article? Support me on Patreon and get direct help from me via a private Slack channel or 1-2-1 mentoring.
Subscribe to my newsletter

Hi, I’m Yan. I’m an AWS Serverless Hero and the author of Production-Ready Serverless.

I specialise in rapidly transitioning teams to serverless and building production-ready services on AWS.

Are you struggling with serverless or need guidance on best practices? Do you want someone to review your architecture and help you avoid costly mistakes down the line? Whatever the case, I’m here to help.

Hire me.

Check out my new course, Complete Guide to AWS Step Functions. 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, callbacks, nested workflows, design patterns and best practices.

Get Your Copy