#### Problem

The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28. In fact, there are exactly four numbers below fifty that can be expressed in such a way:

28 = 2^{2}+ 2^{3}+ 2^{4}

33 = 3^{2}+ 2^{3}+ 2^{4}

49 = 5^{2}+ 2^{3}+ 2^{4}

47 = 2^{2}+ 3^{3}+ 2^{4}

How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?

#### Solution

// generate all prime numbers under <= this max let max = int64(sqrt(double(50000000L))) // initialise the list with 2 which is the only even number in the sequence 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] // 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 = 1L then false else not(hasDivisor(n)) let answer = primeNumbers |> Seq.collect (fun n -> primeNumbers |> Seq.map (fun n' -> pown n 2 + pown n' 3) |> Seq.takeWhile (fun sum -> sum < 50000000L)) |> Seq.collect (fun sum -> primeNumbers |> Seq.map (fun n -> sum + pown n 4) |> Seq.takeWhile (fun sum' -> sum' < 50000000L)) |> Seq.distinct |> Seq.length

The biggest prime that can appear in the equation is equals to the square root of 50000000 – 8 – 16, which, incidentally is just over 7000, so the numbers of primes involved is reasonably small which bodes well for a fast solution!

The logic in this solution is otherwise simple, using the cached prime numbers list to first generate numbers (< 50 million) that can be written as the sum of a prime square and prime cube; then for each number see if we can add a prime fourth power and get a sum less than 50 million.

One thing that caught me out initially was the need to search for distinct numbers as some of these numbers do overlap, but otherwise this is a solution which runs happily under 3 seconds.

**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