Improve problem 75 in Python.

python/e075.py

@@ -1,4 +1,5 @@
 from math import sqrt
+from lib_misc import gcd
 
 
 def proper_divisors(n):
@@ -28,16 +29,6 @@ def triples_brute_force(b_max):
     return triples
 
 
-def triples_euclids_formula(m, n):
-    """ Does not return all triples! """
-    assert(m > n)
-    assert(n > 0)
-    a = m * m - n * n
-    b = 2 * m * n
-    c = m * m + n * n
-    return (a, b, c)
-
-
 def dicksons_method(r):
     triples = []
     assert(r % 2 == 0)
@@ -94,7 +85,7 @@ def dicksons_method_efficient(r):
     return perimeters
 
 
-def euler_075():
+def euler_075_dicksons():
     """
     We are using Dickson's method to get all Pythagorean triples.  The
     challenge is that this method requires finding the divisors for a
@@ -130,6 +121,57 @@ def euler_075():
     return one_integer_sided_right_triangle_count
 
 
+def triples_euclids_formula(m, n):
+    """
+    Generates only primitive triples.
+    https://en.wikipedia.org/wiki/Pythagorean_triple#Generating_a_triple
+    """
+    assert(m > n)
+    assert(n > 0)
+    assert(gcd(m, n) == 1)
+    assert(m % 2 == 0 or n % 2 == 0)
+    a = m * m - n * n
+    b = 2 * m * n
+    c = m * m + n * n
+    if a < b:
+        return (a, b, c)
+    else:
+        return (b, a, c)
+
+
+def euler_075():
+    """
+    The range for m was chosen empirically (aka by trial and error).
+    I tried terminating when L > L_max the first time, but there might
+    be combinations for m = m + 1 that yield lower L. I guess the right
+    criteria to return would be if L > L_max and n == 1.
+    This is good enough for now. I still like my other solution, but it
+    is so much slower because finding the divisors is so expensive.
+    """
+    L = 0
+    L_max = 1500000
+    m = 1
+    perimeter_counts = {}
+    for m in range(1, 1000):
+        for n in range(1, m):
+            try:
+                L = sum(triples_euclids_formula(m, n))
+                L_k = L
+                while L_k <= L_max:
+                    try:
+                        perimeter_counts[L_k] += 1
+                    except KeyError:
+                        perimeter_counts[L_k] = 1
+                    L_k += L
+            except AssertionError:
+                pass
+    one_integer_sided_right_triangle_count = 0
+    for perimeter, count in perimeter_counts.items():
+        if count == 1:
+            one_integer_sided_right_triangle_count += 1
+    return one_integer_sided_right_triangle_count
+
+
 if __name__ == "__main__":
     print("e075.py: " + str(euler_075()))
     assert(euler_075() == 161667)