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 


from fractions import gcd
 
 
def pt1(maxperimeter=100):
    '''
# Naive method
    '''
    trips = []
    for a in range(1, maxperimeter):
        aa = a*a
        for b in range(a, maxperimeter-a+1):
            bb = b*b
            for c in range(b, maxperimeter-b-a+1):
                cc = c*c
                if a+b+c > maxperimeter or cc > aa + bb: break
                if aa + bb == cc:
                    trips.append((a,b,c, gcd(a, b) == 1))
    return trips
 
def pytrip(trip=(3,4,5),perim=100, prim=1):
    a0, b0, c0 = a, b, c = sorted(trip)
    t, firstprim = set(), prim>0
    while a + b + c <= perim:
        t.add((a, b, c, firstprim>0))
        a, b, c, firstprim = a+a0, b+b0, c+c0, False
    #
    t2 = set()
    for a, b, c, firstprim in t:
        a2, a5, b2, b5, c2, c3, c7 = a*2, a*5, b*2, b*5, c*2, c*3, c*7
        if  a5 - b5 + c7 <= perim:
            t2 |= pytrip(( a - b2 + c2,  a2 - b + c2,  a2 - b2 + c3), perim, firstprim)
        if  a5 + b5 + c7 <= perim:
            t2 |= pytrip(( a + b2 + c2,  a2 + b + c2,  a2 + b2 + c3), perim, firstprim)
        if -a5 + b5 + c7 <= perim:
            t2 |= pytrip((-a + b2 + c2, -a2 + b + c2, -a2 + b2 + c3), perim, firstprim)
    return t | t2
 
def pt2(maxperimeter=100):
    '''
# Parent/child relationship method:
# http://en.wikipedia.org/wiki/Formulas_for_generating_Pythagorean_triples#XI.
    '''
    trips = pytrip((3,4,5), maxperimeter, 1)
    return trips
 
def printit(maxperimeter=100, pt=pt1):
    trips = pt(maxperimeter)
    print("  Up to a perimeter of %i there are %i triples, of which %i are primitive"
          % (maxperimeter,
             len(trips),
             len([prim for a,b,c,prim in trips if prim])))
 
for algo, mn, mx in ((pt1, 250, 2500), (pt2, 500, 20000)):
    print(algo.__doc__)
    for maxperimeter in range(mn, mx+1, mn):
        printit(maxperimeter, algo)
 