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 


with Ada.Text_IO;
 
procedure Pythagorean_Triples is
 
   type Large_Natural is range 0 .. 2**63-1;
     -- this is the maximum for gnat
 
   procedure New_Triangle(A, B, C: Large_Natural;
                          Max_Perimeter: Large_Natural;
                          Total_Cnt, Primitive_Cnt: in out Large_Natural) is
      Perimeter: constant Large_Natural := A + B + C;
   begin
      if Perimeter <= Max_Perimeter then
         Primitive_Cnt := Primitive_Cnt + 1;
         Total_Cnt     := Total_Cnt + Max_Perimeter / Perimeter;
         New_Triangle(A-2*B+2*C,     2*A-B+2*C,    2*A-2*B+3*C,   Max_Perimeter, Total_Cnt, Primitive_Cnt);
         New_Triangle(A+2*B+2*C,     2*A+B+2*C,    2*A+2*B+3*C,   Max_Perimeter, Total_Cnt, Primitive_Cnt);
         New_Triangle(2*B+2*C-A,     B+2*C-2*A,    2*B+3*C-2*A,   Max_Perimeter, Total_Cnt, Primitive_Cnt);
      end if;
   end New_Triangle;
 
   T_Cnt, P_Cnt: Large_Natural;
 
begin
   for I in 1 .. 9 loop
      T_Cnt := 0;
      P_Cnt := 0;
      New_Triangle(3,4,5, 10**I, Total_Cnt => T_Cnt, Primitive_Cnt => P_Cnt);
      Ada.Text_IO.Put_Line("Up to 10 **" & Integer'Image(I) & " :" &
                             Large_Natural'Image(T_Cnt) & " Triples," &
                             Large_Natural'Image(P_Cnt) & " Primitives");
   end loop;
end Pythagorean_Triples;