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 


const u = [[ 1, -2,  2,  2, -1,  2,  2, -2,  3],
           [ 1,  2,  2,  2,  1,  2,  2,  2,  3],
           [-1,  2,  2, -2,  1,  2, -2,  2,  3]]
 
var
  total, prim = 0
  maxPeri = 10
 
proc newTri(ins: array[0..2, int]) =
  var p = ins[0] + ins[1] + ins[2]
  if p > maxPeri: return
  inc(prim)
  total += maxPeri div p
 
  for i in 0..2:
    newTri([u[i][0] * ins[0] + u[i][1] * ins[1] + u[i][2] * ins[2],
            u[i][3] * ins[0] + u[i][4] * ins[1] + u[i][5] * ins[2],
            u[i][6] * ins[0] + u[i][7] * ins[1] + u[i][8] * ins[2]])
 
while maxPeri <= 100_000_000:
  total = 0
  prim = 0
  newTri([3, 4, 5])
  echo "Up to ", maxPeri, ": ", total, " triples, ", prim, " primitives"
  maxPeri *= 10