#### Problem

Euler published the remarkable quadratic formula:

n² + n + 41

It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39. However, when n = 40, 40^{2}+ 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n = 41, 41² + 41 + 41 is clearly divisible by 41.

Using computers, the incredible formula n² – 79n + 1601 was discovered, which produces 80 primes for the consecutive values n = 0 to 79. The product of the coefficients, -79 and 1601, is -126479.

Considering quadratics of the form:

n² + an + b, where |a| < 1000 and |b| < 1000

where |n| is the modulus/absolute value of n

e.g. |11| = 11 and |-4| = 4

Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0.

#### Solution

let hasDivisor(n) = let upperBound = int64(sqrt(double(n))) [2L..upperBound] |> Seq.exists (fun x -> n % x = 0L) // need to consider negative values let isPrime(n) = if n <= 1L then false else not(hasDivisor(n)) // the quadratic expression let F (n:int64) (a:int64) (b:int64) = (n*n) + (a*n) + b // function to return the number of consecutive primes the coefficients generate let primeCount a b = Seq.unfold (fun state -> Some(state, state + 1L)) 0L |> Seq.takeWhile (fun n -> isPrime (F n a b)) |> Seq.length let aList, bList = [-999L..999L], [2L..999L] |> List.filter isPrime let answer = let (a, b, _) = aList |> List.collect (fun a -> bList |> List.filter (fun b -> a + b >= 1L) |> List.map (fun b -> (a, b, primeCount a b))) |> List.maxBy (fun (_, _, count) -> count) a * b

Whilst I started off using a brute force approach (which took quite a while to run) I realised that most of the combinations of *a* and *b* don’t generate any primes at all. In order to reduce the number of combinations you need to check, consider the cases when *n = 0* and *n = 1*:

*n = 0*: expression evaluates to *b*, in order for this to generate a prime* b* must be *>= 2*.

*n = 1*: expression evaluates to *1 + a + b*, in order for this to generate a prime *1 + a + b* must be *>= 2*, therefore *a + b >= 1*.

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.

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.

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