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 


$ include "seed7_05.s7i";
  include "bigint.s7i";
 
var bigInteger: total is 0_;
var bigInteger: prim is 0_;
var bigInteger: max_peri is 10_;
 
const proc: new_tri (in bigInteger: a, in bigInteger: b, in bigInteger: c) is func
  local
    var bigInteger: p is 0_;
  begin
    p := a + b + c;
    if p <= max_peri then
      incr(prim);
      total +:= max_peri div p;
      new_tri( a - 2_*b + 2_*c,  2_*a - b + 2_*c,  2_*a - 2_*b + 3_*c);
      new_tri( a + 2_*b + 2_*c,  2_*a + b + 2_*c,  2_*a + 2_*b + 3_*c);
      new_tri(-a + 2_*b + 2_*c, -2_*a + b + 2_*c, -2_*a + 2_*b + 3_*c);
    end if;
  end func;
 
const proc: main is func
  begin
    while max_peri <= 100000000_ do
      total := 0_;
      prim := 0_;
      new_tri(3_, 4_, 5_);
      writeln("Up to " <& max_peri <& ": " <& total <& " triples, " <& prim <& " primitives.");
      max_peri *:= 10_;
    end while;
  end func;