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 


defmodule RC do
  def count_triples(limit), do: count_triples(limit,3,4,5)
 
  defp count_triples(limit, a, b, c) when limit<(a+b+c), do: {0,0}
  defp count_triples(limit, a, b, c) do
    {p1, t1} = count_triples(limit, a-2*b+2*c, 2*a-b+2*c, 2*a-2*b+3*c)
    {p2, t2} = count_triples(limit, a+2*b+2*c, 2*a+b+2*c, 2*a+2*b+3*c)
    {p3, t3} = count_triples(limit,-a+2*b+2*c,-2*a+b+2*c,-2*a+2*b+3*c)
    {1+p1+p2+p3, div(limit, a+b+c)+t1+t2+t3}
  end
end
 
list = for n <- 1..8, do: Enum.reduce(1..n, 1, fn(_,acc)->10*acc end)
Enum.each(list, fn n -> IO.inspect {n, RC.count_triples(n)} end)