#### Problem

By replacing the 1^{st}digit of *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.

By replacing the 3^{rd}and 4^{th}digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers, yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993. Consequently 56003, being the first member of this family, is the smallest prime with this property.

Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.

#### Solution

// generate all prime numbers under <= this max let max = 200000L let mutable primeNumbers = [2L] // only check the prime numbers which are <= the square root of the number n let hasDivisor n = primeNumbers |> Seq.takeWhile (fun n' -> n' <= int64(sqrt(double(n)))) |> Seq.exists (fun n' -> n % n' = 0L) // only check odd numbers <= max let potentialPrimes = Seq.unfold (fun n -> if n > max then None else Some(n, n+2L)) 3L // populate the prime numbers list for n in potentialPrimes do if not(hasDivisor n) then primeNumbers <- primeNumbers @ [n] let isPrime n = if n = 1L then false else not(hasDivisor(n)) // define function to generate combinations of n elements out of the specified list let rec comb n l = match n, l with | 0, _ -> [[]] | _, [] -> [] | k, (x::xs) -> List.map ((@) [x]) (comb (k-1) xs) @ comb k xs // define function to find the wild card digits of a n-digit number let getWildCardDigits n = [1..n-1] |> List.collect (fun n' -> comb n' [0..n-1]) // define function to get the new numbers you get by replacing the digits in the // supplied number n with the same digit let replaceDigits (digits:int list) n = let nDigits = n.ToString().ToCharArray() [0..9] |> List.map (fun n' -> List.init nDigits.Length (fun d -> if List.exists (fun d' -> d' = d) digits then n'.ToString() else nDigits.[d].ToString()) |> List.reduce (+)) |> List.map (int64) // define function to find the lists (if any) of prime numbers obtained by replacing // 1 or more digits of the supplied number n with the same digit let F len n = getWildCardDigits (n.ToString().Length) |> List.map (fun l -> replaceDigits l n |> List.filter (fun n' -> n'.ToString().Length = n.ToString().Length && isPrime n')) |> List.filter (fun l -> l.Length >= len) let answer = primeNumbers |> Seq.skipWhile (fun n -> n < 56003L) |> Seq.map (F 8 ) |> Seq.filter (fun l -> l.Length > 0) |> Seq.head |> Seq.map (fun l -> List.min l) |> Seq.min

**Yan Cui**

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.

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.

- 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