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 


object PythagoreanTriples extends App {
 
  println("               Limit Primatives          All")
 
  for {e <- 2 to 7
       limit = math.pow(10, e).longValue()
  } {
    var primCount, tripCount = 0
 
    def parChild(a: BigInt, b: BigInt, c: BigInt): Unit = {
      val perim = a + b + c
      val (a2, b2, c2, c3) = (2 * a, 2 * b, 2 * c, 3 * c)
      if (limit >= perim) {
        primCount += 1
        tripCount += (limit / perim).toInt
        parChild(a - b2 + c2, a2 - b + c2, a2 - b2 + c3)
        parChild(a + b2 + c2, a2 + b + c2, a2 + b2 + c3)
        parChild(-a + b2 + c2, -a2 + b + c2, -a2 + b2 + c3)
      }
    }
 
    parChild(BigInt(3), BigInt(4), BigInt(5))
    println(f"a + b + c <= ${limit.toFloat}%3.1e  $primCount%9d $tripCount%12d")
  }
}