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#### Problem

Euler’s Totient function, ?(n) [sometimes called the phi function], is used to determine the number of positive numbers less than or equal tonwhich are relatively prime ton. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, ?(9)=6.

The number 1 is considered to be relatively prime to every positive number, so ?(1)=1.

Interestingly, ?(87109)=79180, and it can be seen that 87109 is a permutation of 79180.

Find the value ofn, 1 <n< 10^{7}, for which ?(n) is a permutation ofnand the ration/?(n) produces a minimum.

#### Solution

// generate all prime numbers under <= this max let max = 10000I // initialise the list with 2 which is the only even number in the sequence let mutable primeNumbers = [2I] // only check the prime numbers which are <= the square root of the number n let hasDivisor n = primeNumbers |> Seq.takeWhile (fun n' -> n' <= bigint(sqrt(double(n)))) |> Seq.exists (fun n' -> n % n' = 0I) // only check odd numbers <= max let potentialPrimes = Seq.unfold (fun n -> if n > max then None else Some(n, n+2I)) 3I // populate the prime numbers list for n in potentialPrimes do if not(hasDivisor n) then primeNumbers <- primeNumbers @ [n] // use the same hasDivisor function instead of the prime numbers list as it offers // far greater coverage as the number n is square rooted so this function can // provide a valid test up to max*max let isPrime n = if n = 1I then false else not(hasDivisor(n)) // define function to find the prime denominators for a number n let getPrimeFactors n = let rec getPrimeFactorsRec denominators n = if n = 1I then denominators else let denominator = primeNumbers |> Seq.filter (fun x -> n % x = 0I) |> Seq.head getPrimeFactorsRec (denominators @ [denominator]) (n/denominator) getPrimeFactorsRec [] n // define Euler's totient function let totient n = if n = 1I then 1I else if isPrime n then n-1I else let primeFactors = getPrimeFactors n |> Seq.distinct n * (primeFactors |> Seq.map (fun n' -> n'-1I) |> Seq.reduce (*)) / (primeFactors |> Seq.reduce (*)) // define function to check if two numbers are permutations of each other let isPermutation a b = let aArray = a.ToString().ToCharArray() |> Array.sort let bArray = b.ToString().ToCharArray() |> Array.sort if Array.length aArray <> Array.length bArray then false else Array.forall2 (fun aChar bChar -> aChar = bChar) aArray bArray // check semi-primes less than 10 million let answer = primeNumbers |> Seq.collect (fun n -> primeNumbers |> Seq.filter (fun n' -> n' > n) |> Seq.map (fun n' -> n * n')) |> Seq.filter (fun n' -> n' > 8000000I && n' < 10000000I) |> Seq.map (fun n -> (n, totient n)) |> Seq.filter (fun (n, n') -> isPermutation n n') |> Seq.minBy (fun (n, n') -> double(n) / double(n'))

According to MathWorld the totient function of *n* equals *n – 1*if *n* is a prime, under normal circumstances *n* divided by *totient n* is minimal if *n* is a prime, but then they wouldn’t be permutations of each other as *n* cannot be the permutation of *n – 1*.

A semi-prime on the other hand, *n*, where *p* and *q* are primes and *n = pq*, its totient function equals *(p – 1)(q – 1)*, and therefore:

and is most likely the source of the answer.

The solution here generates all the semi-primes under 10 million and looks for the semi-prime n which returns the smallest* n/totient n*. The larger *n* is the smaller *n/totient n* is, hence to reduce the amount of computation required I only check for numbers greater than 8 million.

Hi, I’m **Yan**. I’m an **AWS Serverless Hero** and I help companies go faster for less by adopting serverless technologies successfully.

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.

### Skill up your serverless game with this hands-on workshop.

My 4-week Production-Ready Serverless online workshop is back!

This course takes you through building a production-ready serverless web application from testing, deployment, security, all the way through to observability. The motivation for this course is to give you **hands-on** experience building something with serverless technologies while giving you a broader view of the challenges you will face as the architecture matures and expands.

We will start at the basics and give you a firm introduction to Lambda and all the relevant concepts and service features (including the latest announcements in 2020). And then gradually ramping up and cover a wide array of topics such as API security, testing strategies, CI/CD, secret management, and operational best practices for monitoring and troubleshooting.

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Check out my new podcast **Real-World Serverless** where I talk with engineers who are building amazing things with serverless technologies and discuss the real-world use cases and challenges they face. If you’re interested in what people are actually doing with serverless and what it’s really like to be working with serverless day-to-day, then this is the podcast for you.

Check out my new course, **Learn you some Lambda best practice for great good!** In this course, you will learn best practices for working with AWS Lambda in terms of performance, cost, security, scalability, resilience and observability. We will also cover latest features from re:Invent 2019 such as Provisioned Concurrency and Lambda Destinations. **Enrol now and start learning**!

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Further reading

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.

- All you need to know about caching for serverless applications
- Lambda optimization tip – enable HTTP keep-alive
- You are wrong about serverless and vendor lock-in
- You are thinking about serverless costs all wrong
- Just how expensive is the full AWS SDK?
- Check-list for going live with API Gateway and Lambda
- How to choose the right API Gateway auth method
- CloudFormation protip: use !Sub instead of !Join
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- Guys, we’re doing pagination wrong
- Top 10 Serverless framework best practices
- How to break the “senior engineer” career ceiling
- My advice to junior developers