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 


Int64 nTriples, nPrimitives, limit
 
F countTriples(Int64 =x, =y, =z)
   L
      V p = x + y + z
      I p > :limit
         R
 
      :nPrimitives++
      :nTriples += :limit I/ p
 
      V t0 = x - 2 * y + 2 * z
      V t1 = 2 * x - y + 2 * z
      V t2 = t1 - y + z
      countTriples(t0, t1, t2)
 
      t0 += 4 * y
      t1 += 2 * y
      t2 += 4 * y
      countTriples(t0, t1, t2)
 
      z = t2 - 4 * x
      y = t1 - 4 * x
      x = t0 - 2 * x
 
L(p) 1..8
   limit = Int64(10) ^ p
   nTriples = nPrimitives = 0
   countTriples(3, 4, 5)
   print(‘Up to #11: #11 triples, #9 primitives.’.format(limit, nTriples, nPrimitives))