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 


Dim total As Variant, prim As Variant, maxPeri As Variant
Private Sub newTri(s0 As Variant, s1 As Variant, s2 As Variant)
    Dim p As Variant
    p = CDec(s0) + CDec(s1) + CDec(s2)
    If p <= maxPeri Then
        prim = prim + 1
        total = total + maxPeri \ p
        newTri s0 + 2 * (-s1 + s2), 2 * (s0 + s2) - s1, 2 * (s0 - s1 + s2) + s2
        newTri s0 + 2 * (s1 + s2), 2 * (s0 + s2) + s1, 2 * (s0 + s1 + s2) + s2
        newTri -s0 + 2 * (s1 + s2), 2 * (-s0 + s2) + s1, 2 * (-s0 + s1 + s2) + s2
      End If
End Sub
Public Sub Program_PythagoreanTriples()
    maxPeri = CDec(100)
    Do While maxPeri <= 10000000#
        prim = CDec(0)
        total = CDec(0)
        newTri 3, 4, 5
        Debug.Print "Up to "; maxPeri; ": "; total; " triples, "; prim; " primitives."
        maxPeri = maxPeri * 10
    Loop
End Sub