Solve problem 853 (getting the remaining 5% once).

python/e853.py

@@ -0,0 +1,40 @@
+def fib():
+    a, b = 1, 1
+    yield a
+    yield b
+    while True:
+        a, b = b, a + b
+        yield b
+
+
+def pisano_period(fibs, n):
+    for period in range(2, 130):
+        if fibs[0] % n == fibs[period] % n and fibs[1] % n == fibs[period + 1] % n:
+            return period
+    return None
+
+
+def euler_853():
+    r = 0
+    fs = fib()
+    fibs = [next(fs) for _ in range(400)]
+    upper = 10**9
+    pi_target = 120
+    for n in range(2, upper + 1):
+        # Check if pi_target is a period.
+        if fibs[0] % n == fibs[pi_target] % n and fibs[1] % n == fibs[pi_target + 1] % n:
+            # Make sure that no lower period than pi_target exits.
+            if pisano_period(fibs, n) == pi_target:
+                r += n
+                # print(n, prime_factors(n))
+                # I have noticed that the highest prime factor is always 61 or
+                # 2521, so we could probably also get the solution by
+                # enumerating the prime factors that yield a sum smaller than
+                # upper and have 61 or 2521 as their last factor.
+    return r
+
+
+if __name__ == "__main__":
+    solution = euler_853()
+    print("e853.py: " + str(solution))
+    assert solution == 44511058204