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 


class PythagoranTriplesCounter
  def initialize(limit = 0)
    @limit = limit
    @total = 0
    @primitives = 0
    generate_triples(3, 4, 5)
  end
 
  def total; @total end
  def primitives; @primitives end
 
  private def generate_triples(a, b, c)
    perim = a + b + c
    return if perim > @limit
 
    @primitives += 1
    @total += @limit // perim
 
    generate_triples( a-2*b+2*c, 2*a-b+2*c, 2*a-2*b+3*c )
    generate_triples( a+2*b+2*c, 2*a+b+2*c, 2*a+2*b+3*c )
    generate_triples(-a+2*b+2*c,-2*a+b+2*c,-2*a+2*b+3*c )
  end
end
 
perim = 10
while perim <= 100_000_000 
  c = PythagoranTriplesCounter.new perim
  p [perim, c.total, c.primitives]
  perim *= 10
end