### The Problem

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

- 9 = 7 + 2×1
^{2}
- 15 = 7 + 2×2
^{2}
- 21 = 3 + 2×3
^{2}
- 25 = 7 + 2×3
^{2}
- 27 = 19 + 2×2
^{2}
- 33 = 31 + 2×1
^{2}

It turns out that the conjecture was false.

What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

### The Solution

As we do with some many of these problems, we need some primes. The problem
definition asks us to find *odd composite numbers* that meet the criteria. In this case
"odd composite" is simply a way of saying "the multiple of two primes other than 2".

We'll start by stealing the `getSomePrimes`

function from
Problem 7.
This function generates a specified number of primes.

```
let intsqrt i = int(sqrt(float i))
let isPrime i =
if i <= 1 then false
elif i = 2 then true
elif (i &&& 1) = 0 then false
else
let sr = intsqrt i
seq { 3..2..sr } |> Seq.forall (fun f -> i%f<>0)
let rec nextPrime i =
match i with
| i when isPrime i -> i
| i -> nextPrime (i + 1)
let getSomePrimes num =
let primeGenerator (candidate,count) =
if count > num then
None
elif candidate = 2 then
Some(2,(3,count+1))
else
let next = nextPrime candidate
Some(next,(next+2,count+1))
Seq.unfold primeGenerator (2,1)
```

The plan of attack is to multiply primes together to generate the composites.
We also need a list of primes to add squares to to test for solutions.
How many primes do we need? I have no idea and lack the mathematical analysis
capabilities to figure it out. So I think 1000 is a good place to start. We can
adjust later if we need to. Because computing primes is expensive, I'm storing them
in a `List`

so they only get generated once.

```
let primes =
getSomePrimes 1000
|> Seq.skip 1
|> List.ofSeq
```

`primes`

will contain a list of 999 primes. It skips over 2 (the first element
in the list) so that we don't generate any even numbers.

Now we need some squares to add to our `primes`

.

```
let generateSquares state =
Some(state * state, state + 1)
let squares =
Seq.unfold generateSquares 1
|> Seq.take 1000
|> List.ofSeq
```

We use `Seq.unfold`

to generate squares for the first 1000 integers. Again, 1000
is just a number I picked to get started. We can adjust later if we need to.

Now we need a way to test possible solutions:

```
let inline isMatch p s n = p + s*2 = n
let findSquare p n =
squares
|> Seq.filter (fun s -> isMatch p s n)
|> Seq.tryHead
let solution n =
primes
|> Seq.filter (fun p -> p < n)
|> Seq.map (fun p -> (p,findSquare p n))
|> Seq.filter (fun (_,square) -> square.IsSome)
|> Seq.tryHead
```

The `isMatch`

function takes a prime `p`

, a square `s`

, and a composite number `n`

as
input. If these inputs satisfy the conjecture, it returns `true`

.

`findSquare`

takes a prime `p`

and a composite number `n`

as input. It returns
the first square that satisfies the conjecture; otherwise it returns `None`

.

The `solution`

function takes a composite number `n`

as input. It takes our list
of `primes`

, filters out any below `n`

, and tries to `findSquare`

that satisfies
the conjecture with that prime. If a solution is found, it returns the solution;
otherwise it returns `None`

.

Now we can solve the problem:

```
let crossJoin xs ys =
xs
|> Seq.collect (fun x -> ys |> Seq.map (fun y -> x, y))
#time
crossJoin primes primes
|> Seq.map (fun (a,b) -> a*b)
|> Seq.sort
|> Seq.map (fun n -> (n, solution n))
|> Seq.filter (fun (n,solution) -> solution.IsNone)
|> Seq.head
|> fst
#time
```

We start by cross joining all our `primes`

with each other prime, multiplying them
together to get composite numbers, and then sorting the resulting sequence. The sort
step is necessary because we are looking for the *smallest* composite that fails
to satisfy the conjecture.

Next, we use `Seq.map`

to generate a solution for each composite. Then we
filter out composites where there is no solution. The head of the resulting
sequence is the answer.

Note: Above, we arbitrarily chose 1000 primes and 1000 squares. It turns out this
generates enough data to find the answer. If it didn't, we'd have to experiment with
increasing these numbers until we finally found the answer.