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 


N=  100;
  a = 1:N;
  b = a(ones(N,1),:).^2;
  b = b+b';
  b = sqrt(b);  [y,x]=find(b==fix(b)); % test 
  % here some alternative tests
  % b = b.^(1/k); [y,x]=find(b==fix(b)); % test 2
  % [y,x]=find(b==(fix(b.^(1/k)).^k));  % test 3
  % b=b.^(1/k); [y,x]=find(abs(b - round(b)) <= 4*eps*b);
 
  z  = sqrt(x.^2+y.^2);
  ix = (z+x+y<100) & (x < y) & (y < z); 
  p  = find(gcd(x(ix),y(ix))==1);   % find primitive triples
 
  printf('There are %i Pythagorean Triples and %i primitive triples with a perimeter smaller than %i.\n',...
         sum(ix), length(p), N);  