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#### 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