diff --git a/Lib/decimal.py b/Lib/decimal.py
index cc71cd8..29ce398 100644
--- a/Lib/decimal.py
+++ b/Lib/decimal.py
@@ -862,7 +862,7 @@
     # that specified by IEEE 754.
 
     def __eq__(self, other, context=None):
-        other = _convert_other(other, allow_float=True)
+        other = _convert_other(other, allow_float = True)
         if other is NotImplemented:
             return other
         if self._check_nans(other, context):
@@ -870,7 +870,7 @@
         return self._cmp(other) == 0
 
     def __ne__(self, other, context=None):
-        other = _convert_other(other, allow_float=True)
+        other = _convert_other(other, allow_float = True)
         if other is NotImplemented:
             return other
         if self._check_nans(other, context):
@@ -879,7 +879,7 @@
 
 
     def __lt__(self, other, context=None):
-        other = _convert_other(other, allow_float=True)
+        other = _convert_other(other, allow_float = True)
         if other is NotImplemented:
             return other
         ans = self._compare_check_nans(other, context)
@@ -888,7 +888,7 @@
         return self._cmp(other) < 0
 
     def __le__(self, other, context=None):
-        other = _convert_other(other, allow_float=True)
+        other = _convert_other(other, allow_float = True)
         if other is NotImplemented:
             return other
         ans = self._compare_check_nans(other, context)
@@ -897,7 +897,7 @@
         return self._cmp(other) <= 0
 
     def __gt__(self, other, context=None):
-        other = _convert_other(other, allow_float=True)
+        other = _convert_other(other, allow_float = True)
         if other is NotImplemented:
             return other
         ans = self._compare_check_nans(other, context)
@@ -906,7 +906,7 @@
         return self._cmp(other) > 0
 
     def __ge__(self, other, context=None):
-        other = _convert_other(other, allow_float=True)
+        other = _convert_other(other, allow_float = True)
         if other is NotImplemented:
             return other
         ans = self._compare_check_nans(other, context)
@@ -935,55 +935,28 @@
 
     def __hash__(self):
         """x.__hash__() <==> hash(x)"""
-        # Decimal integers must hash the same as the ints
-        #
-        # The hash of a nonspecial noninteger Decimal must depend only
-        # on the value of that Decimal, and not on its representation.
-        # For example: hash(Decimal('100E-1')) == hash(Decimal('10')).
 
-        # Equality comparisons involving signaling nans can raise an
-        # exception; since equality checks are implicitly and
-        # unpredictably used when checking set and dict membership, we
-        # prevent signaling nans from being used as set elements or
-        # dict keys by making __hash__ raise an exception.
+        # In order to make sure that the hash of a Decimal instance
+        # agrees with the hash of a numerically equal integer, float
+        # or Fraction, we follow the rules for numeric hashes outlined
+        # in the documentation.  (See library docs, 'Built-in Types').
         if self._is_special:
             if self.is_snan():
                 raise TypeError('Cannot hash a signaling NaN value.')
             elif self.is_nan():
-                # 0 to match hash(float('nan'))
-                return 0
+                return _PyHASH_NAN
             else:
-                # values chosen to match hash(float('inf')) and
-                # hash(float('-inf')).
                 if self._sign:
-                    return -271828
+                    return -_PyHASH_INF
                 else:
-                    return 314159
+                    return _PyHASH_INF
 
-        # In Python 2.7, we're allowing comparisons (but not
-        # arithmetic operations) between floats and Decimals;  so if
-        # a Decimal instance is exactly representable as a float then
-        # its hash should match that of the float.
-        self_as_float = float(self)
-        if Decimal.from_float(self_as_float) == self:
-            return hash(self_as_float)
-
-        if self._isinteger():
-            op = _WorkRep(self.to_integral_value())
-            # to make computation feasible for Decimals with large
-            # exponent, we use the fact that hash(n) == hash(m) for
-            # any two nonzero integers n and m such that (i) n and m
-            # have the same sign, and (ii) n is congruent to m modulo
-            # 2**64-1.  So we can replace hash((-1)**s*c*10**e) with
-            # hash((-1)**s*c*pow(10, e, 2**64-1).
-            return hash((-1)**op.sign*op.int*pow(10, op.exp, 2**64-1))
-        # The value of a nonzero nonspecial Decimal instance is
-        # faithfully represented by the triple consisting of its sign,
-        # its adjusted exponent, and its coefficient with trailing
-        # zeros removed.
-        return hash((self._sign,
-                     self._exp+len(self._int),
-                     self._int.rstrip('0')))
+        if self._exp >= 0:
+            exp_hash = pow(10, self._exp, _PyHASH_MODULUS)
+        else:
+            exp_hash = pow(_PyHASH_10INV, -self._exp, _PyHASH_MODULUS)
+        hash_ = int(self._int) * exp_hash % _PyHASH_MODULUS
+        return hash_ if self >= 0 else -hash_
 
     def as_tuple(self):
         """Represents the number as a triple tuple.
@@ -6218,6 +6191,17 @@
 # _SignedInfinity[sign] is infinity w/ that sign
 _SignedInfinity = (_Infinity, _NegativeInfinity)
 
+# Constants related to the hash implementation;  hash(x) is based
+# on the reduction of x modulo _PyHASH_MODULUS
+import sys
+_PyHASH_MODULUS = sys.hash_info.modulus
+# hash values to use for positive and negative infinities, and nans
+_PyHASH_INF = sys.hash_info.inf
+_PyHASH_NAN = sys.hash_info.nan
+del sys
+
+# _PyHASH_10INV is the inverse of 10 modulo the prime _PyHASH_MODULUS
+_PyHASH_10INV = pow(10, _PyHASH_MODULUS - 2, _PyHASH_MODULUS)
 
 
 if __name__ == '__main__':
diff --git a/Lib/fractions.py b/Lib/fractions.py
index fc8a12c..51e67e2 100644
--- a/Lib/fractions.py
+++ b/Lib/fractions.py
@@ -8,6 +8,7 @@
 import numbers
 import operator
 import re
+import sys
 
 __all__ = ['Fraction', 'gcd']
 
@@ -23,6 +24,12 @@
         a, b = b, a%b
     return a
 
+# Constants related to the hash implementation;  hash(x) is based
+# on the reduction of x modulo the prime _PyHASH_MODULUS.
+_PyHASH_MODULUS = sys.hash_info.modulus
+# Value to be used for rationals that reduce to infinity modulo
+# _PyHASH_MODULUS.
+_PyHASH_INF = sys.hash_info.inf
 
 _RATIONAL_FORMAT = re.compile(r"""
     \A\s*                      # optional whitespace at the start, then
@@ -528,16 +535,22 @@
 
         """
         # XXX since this method is expensive, consider caching the result
-        if self._denominator == 1:
-            # Get integers right.
-            return hash(self._numerator)
-        # Expensive check, but definitely correct.
-        if self == float(self):
-            return hash(float(self))
+
+        # In order to make sure that the hash of a Fraction agrees
+        # with the hash of a numerically equal integer, float or
+        # Decimal instance, we follow the rules for numeric hashes
+        # outlined in the documentation.  (See library docs, 'Built-in
+        # Types').
+
+        # dinv is the inverse of self._denominator modulo the prime
+        # _PyHASH_MODULUS, or 0 if self._denominator is divisible by
+        # _PyHASH_MODULUS.
+        dinv = pow(self._denominator, _PyHASH_MODULUS - 2, _PyHASH_MODULUS)
+        if not dinv:
+            hash_ = _PyHASH_INF
         else:
-            # Use tuple's hash to avoid a high collision rate on
-            # simple fractions.
-            return hash((self._numerator, self._denominator))
+            hash_ = abs(self._numerator) * dinv % _PyHASH_MODULUS
+        return hash_ if self >= 0 else -hash_
 
     def __eq__(a, b):
         """a == b"""
diff --git a/Lib/test/test_float.py b/Lib/test/test_float.py
index b52b1db..cabeb16 100644
--- a/Lib/test/test_float.py
+++ b/Lib/test/test_float.py
@@ -914,15 +914,6 @@
         self.assertFalse(NAN.is_inf())
         self.assertFalse((0.).is_inf())
 
-    def test_hash_inf(self):
-        # the actual values here should be regarded as an
-        # implementation detail, but they need to be
-        # identical to those used in the Decimal module.
-        self.assertEqual(hash(float('inf')), 314159)
-        self.assertEqual(hash(float('-inf')), -271828)
-        self.assertEqual(hash(float('nan')), 0)
-
-
 fromHex = float.fromhex
 toHex = float.hex
 class HexFloatTestCase(unittest.TestCase):
diff --git a/Lib/test/test_numeric_tower.py b/Lib/test/test_numeric_tower.py
new file mode 100644
index 0000000..eafdb0f
--- /dev/null
+++ b/Lib/test/test_numeric_tower.py
@@ -0,0 +1,151 @@
+# test interactions betwen int, float, Decimal and Fraction
+
+import unittest
+import random
+import math
+import sys
+import operator
+from test.support import run_unittest
+
+from decimal import Decimal as D
+from fractions import Fraction as F
+
+# Constants related to the hash implementation;  hash(x) is based
+# on the reduction of x modulo the prime _PyHASH_MODULUS.
+_PyHASH_MODULUS = sys.hash_info.modulus
+_PyHASH_INF = sys.hash_info.inf
+
+class HashTest(unittest.TestCase):
+    def check_equal_hash(self, x, y):
+        # check both that x and y are equal and that their hashes are equal
+        self.assertEqual(hash(x), hash(y),
+                         "got different hashes for {!r} and {!r}".format(x, y))
+        self.assertEqual(x, y)
+
+    def test_bools(self):
+        self.check_equal_hash(False, 0)
+        self.check_equal_hash(True, 1)
+
+    def test_integers(self):
+        # check that equal values hash equal
+
+        # exact integers
+        for i in range(-1000, 1000):
+            self.check_equal_hash(i, float(i))
+            self.check_equal_hash(i, D(i))
+            self.check_equal_hash(i, F(i))
+
+        # the current hash is based on reduction modulo 2**n-1 for some
+        # n, so pay special attention to numbers of the form 2**n and 2**n-1.
+        for i in range(100):
+            n = 2**i - 1
+            if n == int(float(n)):
+                self.check_equal_hash(n, float(n))
+                self.check_equal_hash(-n, -float(n))
+            self.check_equal_hash(n, D(n))
+            self.check_equal_hash(n, F(n))
+            self.check_equal_hash(-n, D(-n))
+            self.check_equal_hash(-n, F(-n))
+
+            n = 2**i
+            self.check_equal_hash(n, float(n))
+            self.check_equal_hash(-n, -float(n))
+            self.check_equal_hash(n, D(n))
+            self.check_equal_hash(n, F(n))
+            self.check_equal_hash(-n, D(-n))
+            self.check_equal_hash(-n, F(-n))
+
+        # random values of various sizes
+        for _ in range(1000):
+            e = random.randrange(300)
+            n = random.randrange(-10**e, 10**e)
+            self.check_equal_hash(n, D(n))
+            self.check_equal_hash(n, F(n))
+            if n == int(float(n)):
+                self.check_equal_hash(n, float(n))
+
+    def test_binary_floats(self):
+        # check that floats hash equal to corresponding Fractions and Decimals
+
+        # floats that are distinct but numerically equal should hash the same
+        self.check_equal_hash(0.0, -0.0)
+
+        # zeros
+        self.check_equal_hash(0.0, D(0))
+        self.check_equal_hash(-0.0, D(0))
+        self.check_equal_hash(-0.0, D('-0.0'))
+        self.check_equal_hash(0.0, F(0))
+
+        # infinities and nans
+        self.check_equal_hash(float('inf'), D('inf'))
+        self.check_equal_hash(float('-inf'), D('-inf'))
+
+        for _ in range(1000):
+            x = random.random() * math.exp(random.random()*200.0 - 100.0)
+            self.check_equal_hash(x, D.from_float(x))
+            self.check_equal_hash(x, F.from_float(x))
+
+    def test_complex(self):
+        # complex numbers with zero imaginary part should hash equal to
+        # the corresponding float
+
+        test_values = [0.0, -0.0, 1.0, -1.0, 0.40625, -5136.5,
+                       float('inf'), float('-inf')]
+
+        for zero in -0.0, 0.0:
+            for value in test_values:
+                self.check_equal_hash(value, complex(value, zero))
+
+    def test_decimals(self):
+        # check that Decimal instances that have different representations
+        # but equal values give the same hash
+        zeros = ['0', '-0', '0.0', '-0.0e10', '000e-10']
+        for zero in zeros:
+            self.check_equal_hash(D(zero), D(0))
+
+        self.check_equal_hash(D('1.00'), D(1))
+        self.check_equal_hash(D('1.00000'), D(1))
+        self.check_equal_hash(D('-1.00'), D(-1))
+        self.check_equal_hash(D('-1.00000'), D(-1))
+        self.check_equal_hash(D('123e2'), D(12300))
+        self.check_equal_hash(D('1230e1'), D(12300))
+        self.check_equal_hash(D('12300'), D(12300))
+        self.check_equal_hash(D('12300.0'), D(12300))
+        self.check_equal_hash(D('12300.00'), D(12300))
+        self.check_equal_hash(D('12300.000'), D(12300))
+
+    def test_fractions(self):
+        # check special case for fractions where either the numerator
+        # or the denominator is a multiple of _PyHASH_MODULUS
+        self.assertEqual(hash(F(1, _PyHASH_MODULUS)), _PyHASH_INF)
+        self.assertEqual(hash(F(-1, 3*_PyHASH_MODULUS)), -_PyHASH_INF)
+        self.assertEqual(hash(F(7*_PyHASH_MODULUS, 1)), 0)
+        self.assertEqual(hash(F(-_PyHASH_MODULUS, 1)), 0)
+
+    def test_hash_normalization(self):
+        # Test for a bug encountered while changing long_hash.
+        #
+        # Given objects x and y, it should be possible for y's
+        # __hash__ method to return hash(x) in order to ensure that
+        # hash(x) == hash(y).  But hash(x) is not exactly equal to the
+        # result of x.__hash__(): there's some internal normalization
+        # to make sure that the result fits in a C long, and is not
+        # equal to the invalid hash value -1.  This internal
+        # normalization must therefore not change the result of
+        # hash(x) for any x.
+
+        class HalibutProxy:
+            def __hash__(self):
+                return hash('halibut')
+            def __eq__(self, other):
+                return other == 'halibut'
+
+        x = {'halibut', HalibutProxy()}
+        self.assertEqual(len(x), 1)
+
+
+def test_main():
+    run_unittest(HashTest)
+
+if __name__ == '__main__':
+    test_main()
diff --git a/Lib/test/test_sys.py b/Lib/test/test_sys.py
index 2caf09f..c056f9a 100644
--- a/Lib/test/test_sys.py
+++ b/Lib/test/test_sys.py
@@ -426,6 +426,23 @@
         self.assertEqual(type(sys.int_info.bits_per_digit), int)
         self.assertEqual(type(sys.int_info.sizeof_digit), int)
         self.assertIsInstance(sys.hexversion, int)
+
+        self.assertEqual(len(sys.hash_info), 5)
+        self.assertLess(sys.hash_info.modulus, 2**sys.hash_info.width)
+        # sys.hash_info.modulus should be a prime; we do a quick
+        # probable primality test (doesn't exclude the possibility of
+        # a Carmichael number)
+        for x in range(1, 100):
+            self.assertEqual(
+                pow(x, sys.hash_info.modulus-1, sys.hash_info.modulus),
+                1,
+                "sys.hash_info.modulus {} is a non-prime".format(
+                    sys.hash_info.modulus)
+                )
+        self.assertIsInstance(sys.hash_info.inf, int)
+        self.assertIsInstance(sys.hash_info.nan, int)
+        self.assertIsInstance(sys.hash_info.imag, int)
+
         self.assertIsInstance(sys.maxsize, int)
         self.assertIsInstance(sys.maxunicode, int)
         self.assertIsInstance(sys.platform, str)