Problem 026

e026.py

@@ -27,47 +27,71 @@ def coprime(a, b):
     primes = [p for p in primes_a if p in primes_b]
     return bool(primes)
 
-def period_length(numerator, denominator):
+def long_division(dividend, divisor):
+    dividend = str(dividend).replace('.', '')
+    i = -1
+    remainder = 0
+    while True:
+        while remainder < divisor:
+            i += 1
+            remainder *= 10
+            try:
+                remainder += int(dividend[i])
+            except:
+                pass
+            if remainder == 0:
+                return
+        digit = str(remainder // divisor)
+        remainder = remainder % divisor
+        yield digit, remainder
+
+def get_cycle_length(numerator, denominator):
     """Determine the period length of a non-terminating decimal"""
 
     length = None
     numerator = Decimal(numerator)
     denominator = Decimal(denominator)
-    if numerator == 1:
-        if denominator == 1:
-            return 1
-        primes = pfactor(int(denominator))
 
-        """
+    if denominator == 1:
-        From Wikipedia:
+        return 1
-            Terminating decimals represent rational numbers of the form
-            k/2^n5^m
     
-            However, a terminating decimal also has a representation as a
+    """
-            repeating decimal, obtained by decreasing the final (nonzero)
+    From Wikipedia:
-            digit by one and appending an infinitely repeating sequence of
+        Terminating decimals represent rational numbers of the form
-            nines. 1 = 0.999999... and 1.585 = 1.584999999... are two
+        k/2^n5^m
-            examples of this.
+
-        """
+        However, a terminating decimal also has a representation as a
-        if len([p for p in primes if p not in [2, 5]]) == 0:
+        repeating decimal, obtained by decreasing the final (nonzero)
-            return 1
+        digit by one and appending an infinitely repeating sequence of
+        nines. 1 = 0.999999... and 1.585 = 1.584999999... are two
+        examples of this.
+    """
+    if coprime(denominator, 10):
+        return 1
+
+    digits = []
+    for digit in long_division(numerator, denominator):
+        if digit in digits:
+            cycle = digits[digits.index(digit):]
+            return len(cycle)
+        digits.append(digit)
 
-        if coprime(denominator, 10):
-            pass
     else:
         raise Exception('Non-reciprocal period detection not yet implemented')
     return length
 
 def main():
-    MAX = 10
+    MAX = 1000
-    longest_cycle = 0
+    longest_cycle = (0, 0)
     for i in xrange((MAX), 1, -1):
-        # The period of 1/k for integer k is always ² k ? 1.
+        # The period of 1/k for integer k is always <= k - 1
-        if longest_cycle >= i:
+        if longest_cycle[1] >= i:
             break
         
-        cycle = period_length(1, i)
+        cycle = get_cycle_length(1, i)
-        fraction = Decimal(1) / Decimal(i)
+        if longest_cycle[1] < cycle:
-        print '1/{0} = {1} ({2})'.format(i, fraction, cycle), fraction.is_subnormal()
+            longest_cycle = (i, cycle)
+    (i, cycle) = longest_cycle
+    print 'Longest cycle length is', cycle, 'for i =', i
 if __name__ == '__main__':
     main()