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 


// version 1.1.2
 
var total = 0L
var prim = 0L
var maxPeri = 0L
 
fun newTri(s0: Long, s1: Long, s2: Long) {
    val p = s0 + s1 + s2
    if (p <= maxPeri) {
        prim++
        total += maxPeri / p
        newTri( s0 - 2 * s1 + 2 * s2,  2 * s0 - s1 + 2 * s2,  2 * s0 - 2 * s1 + 3 * s2)
        newTri( s0 + 2 * s1 + 2 * s2,  2 * s0 + s1 + 2 * s2,  2 * s0 + 2 * s1 + 3 * s2)
        newTri(-s0 + 2 * s1 + 2 * s2, -2 * s0 + s1 + 2 * s2, -2 * s0 + 2 * s1 + 3 * s2)
    }
}
 
fun main(args: Array<String>) {
    maxPeri = 100
    while (maxPeri <= 10_000_000_000L) {
        prim = 0
        total = 0
        newTri(3, 4, 5)
        println("Up to $maxPeri: $total triples, $prim primatives")
        maxPeri *= 10
    }
}