From 8b1a4fa6ef7f4091544f5d67a02b8a16a22af3fe Mon Sep 17 00:00:00 2001
From: Felix Martin <mail@felixm.de>
Date: Thu, 1 Aug 2019 09:38:51 -0400
Subject: [PATCH] Solve 71 in Python.

---
 python/e071.py     | 30 ++++++++++++++++++++++++++++--
 python/lib_misc.py |  1 +
 2 files changed, 29 insertions(+), 2 deletions(-)

diff --git a/python/e071.py b/python/e071.py
index 49a122e..cbfc096 100644
--- a/python/e071.py
+++ b/python/e071.py
@@ -1,8 +1,34 @@
+from lib_misc import gcd
+
+
+def smallest_fraction_with_new_denominator(n, d, d_new):
+    return (int(d_new / d * n), d_new)
+
 
 def euler_071():
-    return 0
+    """
+    The idea is to find the the closest fraction to (n, d) for every potential
+    new denominator d_new. We do that by extrapolating (n, d) to d_new and
+    rounding n_new down. This is what smallest_fraction_with_new_denominator
+    does.
+
+    We can then check if the greatest common divisor for n_new and d_new is
+    one. I though it might be necessary to search for the next smallest n_new
+    if gcd(n_new, d_new) > 1, but it turned out that this is not actually
+    necessary.
+    """
+    n = 3
+    d = 7
+    n_min, d_min = 0, 1
+    for d_new in range(950000, 1000000):
+        n_new, d_new = smallest_fraction_with_new_denominator(n, d, d_new)
+        if d_new == d:
+            pass
+        elif gcd(n_new, d_new) == 1 and n_new / d_new > n_min / d_min:
+            n_min, d_min = n_new, d_new
+    return n_min
 
 
 if __name__ == "__main__":
     print("e071.py: " + str(euler_071()))
-    assert(euler_071() == 0)
+    assert(euler_071() == 428570)
diff --git a/python/lib_misc.py b/python/lib_misc.py
index 7552b4f..1d13e0e 100644
--- a/python/lib_misc.py
+++ b/python/lib_misc.py
@@ -195,6 +195,7 @@ def permutations(iterable):
             yield elem + ps
 
 
+@lru_cache(maxsize=1000000)
 def gcd(a, b):
     while a % b != 0:
         a, b = b, a % b