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 


      DIM U0%(2,2), U1%(2,2), U2%(2,2), seed%(2)
      U0%() =  1, -2, 2,  2, -1, 2,  2, -2, 3
      U1%() =  1,  2, 2,  2,  1, 2,  2,  2, 3
      U2%() = -1,  2, 2, -2,  1, 2, -2,  2, 3
 
      seed%() = 3, 4, 5
      FOR power% = 1 TO 7
        all% = 0 : prim% = 0
        PROCtri(seed%(), 10^power%, all%, prim%)
        PRINT "Up to 10^"; power%, ": " all% " triples" prim% " primitives"
      NEXT
      END
 
      DEF PROCtri(i%(), mp%, RETURN all%, RETURN prim%)
      LOCAL t%() : DIM t%(2)
 
      IF SUM(i%()) > mp% ENDPROC
      prim% += 1
      all% += mp% DIV SUM(i%())
 
      t%() = U0%() . i%()
      PROCtri(t%(), mp%, all%, prim%)
      t%() = U1%() . i%()
      PROCtri(t%(), mp%, all%, prim%)
      t%() = U2%() . i%()
      PROCtri(t%(), mp%, all%, prim%)
      ENDPROC