Problem Statement

Create a reasonably complete implementation of rational arithmetic in the particular language using the idioms of the language.

Define a new type called frac with binary operator "//" of two integers that returns a structure made up of the numerator and the denominator (as per a rational number). Further define the appropriate rational unary operators abs and '-', with the binary operators for addition '+', subtraction '-', multiplication '×', division '/', integer division '÷', modulo division, the comparison operators (e.g. '<', '≤', '>', & '≥') and equality operators (e.g. '=' & '≠'). Define standard coercion operators for casting int to frac etc. If space allows, define standard increment and decrement operators (e.g. '+:=' & '-:=' etc.). Finally test the operators: Use the new type frac to find all perfect numbers less than 219 by summing the reciprocal of the factors.

Let's start with the solution:

type frac = Microsoft.FSharp.Math.BigRational

let perf n = 1N = List.fold (+) 0N (List.map (fun i -> if n % i = 0 then 1N/frac.FromInt(i) else 0N) [2..n])

for i in 1..(1<<<19) do if (perf i) then printfn "%i is perfect" i