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 


(defn gcd [a b] (if (zero? b) a (recur b (mod a b))))
 
(defn pyth [peri]
  (for [m (range 2 (Math/sqrt (/ peri 2)))
        n (range (inc (mod m 2)) m 2) ; n<m, opposite polarity
        :let [p (* 2 m (+ m n))]      ; = a+b+c for this (m,n)
        :while (<= p peri)
        :when (= 1 (gcd m n))
        :let [m2 (* m m), n2 (* n n), 
              [a b] (sort [(- m2 n2) (* 2 m n)]), c (+ m2 n2)]
        k (range 1 (inc (quot peri p)))]
    [(= k 1) (* k a) (* k b) (* k c)]))
 
(defn rcount [ts] ; (->> peri pyth rcount) produces [total, primitive] counts
  (reduce (fn [[total prims] t] [(inc total), (if (first t) (inc prims) prims)])
    [0 0]
    ts))