A Pythagorean triple is defined as three positive integers 



(
a
,
b
,
c
)


{\displaystyle (a,b,c)}

 where 



a
<
b
<
c


{\displaystyle a<b<c}

, and 




a

2


+

b

2


=

c

2


.


{\displaystyle a^{2}+b^{2}=c^{2}.}



They are called primitive triples if 



a
,
b
,
c


{\displaystyle a,b,c}

 are co-prime, that is, if their pairwise greatest common divisors 





g
c
d


(
a
,
b
)
=


g
c
d


(
a
,
c
)
=


g
c
d


(
b
,
c
)
=
1


{\displaystyle {\rm {gcd}}(a,b)={\rm {gcd}}(a,c)={\rm {gcd}}(b,c)=1}

.

Because of their relationship through the Pythagorean theorem, a, b, and c are co-prime if a and b are co-prime (





g
c
d


(
a
,
b
)
=
1


{\displaystyle {\rm {gcd}}(a,b)=1}

).  

Each triple forms the length of the sides of a right triangle, whose perimeter is 



P
=
a
+
b
+
c


{\displaystyle P=a+b+c}

.



Task

The task is to determine how many Pythagorean triples there are with a perimeter no larger than 100 and the number of these that are primitive.



Extra credit
 
Deal with large values.   Can your program handle a maximum perimeter of 1,000,000?   What about 10,000,000?   100,000,000?

Note: the extra credit is not for you to demonstrate how fast your language is compared to others;   you need a proper algorithm to solve them in a timely manner.



Related tasks

   Euler's sum of powers conjecture 
   List comprehensions
   Pythagorean quadruples 


 
:- module comprehension.
:- interface.
:- import_module io.
:- import_module int.
 
:- type triple ---> triple(int, int, int).
 
:- pred pythTrip(int::in,triple::out) is nondet.
:- pred main(io::di, io::uo) is det.
 
:- implementation.
:- import_module solutions.
 
pythTrip(Limit,triple(X,Y,Z)) :-
    nondet_int_in_range(1,Limit,X),
    nondet_int_in_range(X,Limit,Y),
    nondet_int_in_range(Y,Limit,Z),
    pow(Z,2) = pow(X,2) + pow(Y,2).
 
main(!IO) :-
    solutions((pred(Triple::out) is nondet :- pythTrip(20,Triple)),Result),
    write(Result,!IO).
 