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 


 
For i=1 To 8
	WScript.StdOut.WriteLine triples(10^i)
Next
 
Function triples(pmax)
	prim=0 : count=0 : nmax=Sqr(pmax)/2 : n=1
	Do While n <= nmax
		m=n+1 : p=2*m*(m+n)
		Do While p <= pmax
			If gcd(m,n)=1 Then
				prim=prim+1
				count=count+Int(pmax/p)
			End If
			m=m+2
			p=2*m*(m+n)
		Loop
		n=n+1
	Loop 
	triples = "Max Perimeter: " & pmax &_
				", Total: " & count &_
				", Primitive: " & prim
End Function
 
Function gcd(a,b)
	c = a : d = b
	Do
		If c Mod d > 0 Then
			e = c Mod d
			c = d
			d = e
		Else
			gcd = d
			Exit Do
		End If
	Loop
End Function
 