(* Euler project Problem 3: Largest prime factor The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ? *) module PrimeFactor use mach.int.Int use number.Divisibility use number.Prime use number.Coprime (** returns the smallest divisor of `n` greater than or equal to `d`, assuming no divisor between 2 and `d`. *) let rec smallest_divisor (d n:int) : int requires { 2 <= n } requires { 2 <= d <= n } requires { forall i:int. 2 <= i < d -> not divides i n } ensures { d <= result <= n } ensures { divides result n } ensures { forall i:int. 2 <= i < result -> not divides i n } variant { n - d } = if d * d > n then begin assert { forall i:int. 2 <= i < n /\ divides i n -> i >= d && let u = div n i in u * i = n && divides u n && u * i = n && (u >= d -> n >= d * i >= d * d && false) && u >= 2 && u < n && false } ; n end else if d >= 2 && n % d = 0 then d else smallest_divisor (d+1) n use ref.Ref use list.List let largest_prime_factor (n:int) : int requires { 2 <= n } ensures { prime result } ensures { divides result n } ensures { forall i:int. result < i <= n -> not (prime i /\ divides i n) } = let d = smallest_divisor 2 n in let factor = ref d in let target = ref (n / d) in assert { !target * d = n && divides !target n } ; assert { forall i:int. prime i /\ divides i n /\ i > d -> coprime d i && divides i !target }; while !target >= 2 do invariant { 1 <= !target <= n } invariant { 2 <= !factor <= n } invariant { divides !factor n } invariant { prime !factor } invariant { forall i:int. divides i !target /\ i >= 2 -> i >= !factor /\ divides i n } invariant { forall i:int. prime i /\ divides i n /\ i > !factor -> divides i !target } variant { !target } let oldt = ghost !target in let ghost oldf = !factor in assert { divides !target !target && !target >= 2 && !target >= !factor }; let d = smallest_divisor !factor !target in assert { prime d }; factor := d; target := !target / d; assert { !target * d = oldt && divides !target oldt } ; assert { forall i:int. prime i /\ divides i n /\ i > d -> i > oldf && divides i oldt && 1 <= d < i && coprime d i && divides i !target } done; !factor let test () = largest_prime_factor 13195 (* should be 29 *) let solve () = largest_prime_factor 600851475143 end (*** Local Variables: compile-command: "why3ide largest_prime_factor.mlw" End: *)