diff --git a/Lib/test/seq_tests.py b/Lib/test/seq_tests.py
index 962cfe1..5148d4b 100644
--- a/Lib/test/seq_tests.py
+++ b/Lib/test/seq_tests.py
@@ -304,11 +304,13 @@
             self.assertEqual(id(s), id(s*1))
 
     def test_bigrepeat(self):
-        x = self.type2test([0])
-        x *= 2**16
-        self.assertRaises(MemoryError, x.__mul__, 2**16)
-        if hasattr(x, '__imul__'):
-            self.assertRaises(MemoryError, x.__imul__, 2**16)
+        import sys
+        if sys.maxsize <= 2147483647:
+            x = self.type2test([0])
+            x *= 2**16
+            self.assertRaises(MemoryError, x.__mul__, 2**16)
+            if hasattr(x, '__imul__'):
+                self.assertRaises(MemoryError, x.__imul__, 2**16)
 
     def test_subscript(self):
         a = self.type2test([10, 11])
diff --git a/Lib/test/support.py b/Lib/test/support.py
index d026984..24aff5e 100644
--- a/Lib/test/support.py
+++ b/Lib/test/support.py
@@ -68,6 +68,7 @@
 use_resources = None     # Flag set to [] by regrtest.py
 max_memuse = 0           # Disable bigmem tests (they will still be run with
                          # small sizes, to make sure they work.)
+real_max_memuse = 0
 
 # _original_stdout is meant to hold stdout at the time regrtest began.
 # This may be "the real" stdout, or IDLE's emulation of stdout, or whatever.
@@ -599,12 +600,14 @@
 _1M = 1024*1024
 _1G = 1024 * _1M
 _2G = 2 * _1G
+_4G = 4 * _1G
 
 MAX_Py_ssize_t = sys.maxsize
 
 def set_memlimit(limit):
     import re
     global max_memuse
+    global real_max_memuse
     sizes = {
         'k': 1024,
         'm': _1M,
@@ -616,6 +619,7 @@
     if m is None:
         raise ValueError('Invalid memory limit %r' % (limit,))
     memlimit = int(float(m.group(1)) * sizes[m.group(3).lower()])
+    real_max_memuse = memlimit
     if memlimit > MAX_Py_ssize_t:
         memlimit = MAX_Py_ssize_t
     if memlimit < _2G - 1:
@@ -661,6 +665,27 @@
         return wrapper
     return decorator
 
+def precisionbigmemtest(size, memuse, overhead=5*_1M):
+    def decorator(f):
+        def wrapper(self):
+            if not real_max_memuse:
+                maxsize = 5147
+            else:
+                maxsize = size
+
+                if real_max_memuse and real_max_memuse < maxsize * memuse:
+                    if verbose:
+                        sys.stderr.write("Skipping %s because of memory "
+                                         "constraint\n" % (f.__name__,))
+                    return
+
+            return f(self, maxsize)
+        wrapper.size = size
+        wrapper.memuse = memuse
+        wrapper.overhead = overhead
+        return wrapper
+    return decorator
+
 def bigaddrspacetest(f):
     """Decorator for tests that fill the address space."""
     def wrapper(self):
diff --git a/Lib/test/test_bigmem.py b/Lib/test/test_bigmem.py
index 30e1bdb..5edc139 100644
--- a/Lib/test/test_bigmem.py
+++ b/Lib/test/test_bigmem.py
@@ -1,5 +1,5 @@
 from test import support
-from test.support import bigmemtest, _1G, _2G
+from test.support import bigmemtest, _1G, _2G, _4G, precisionbigmemtest
 
 import unittest
 import operator
@@ -53,6 +53,22 @@
         self.assertEquals(s[lpadsize:-rpadsize], SUBSTR)
         self.assertEquals(s.strip(), SUBSTR.strip())
 
+    @precisionbigmemtest(size=_2G - 1, memuse=1)
+    def test_center_unicode(self, size):
+        SUBSTR = ' abc def ghi'
+        try:
+            s = SUBSTR.center(size)
+        except OverflowError:
+            pass # acceptable on 32-bit
+        else:
+            self.assertEquals(len(s), size)
+            lpadsize = rpadsize = (len(s) - len(SUBSTR)) // 2
+            if len(s) % 2:
+                lpadsize += 1
+            self.assertEquals(s[lpadsize:-rpadsize], SUBSTR)
+            self.assertEquals(s.strip(), SUBSTR.strip())
+            del s
+
     @bigmemtest(minsize=_2G, memuse=2)
     def test_count(self, size):
         SUBSTR = ' abc def ghi'
@@ -69,10 +85,51 @@
         s = b'.' * size
         self.assertEquals(len(s.decode('utf-8')), size)
 
+    def basic_encode_test(self, size, enc, c='.', expectedsize=None):
+        if expectedsize is None:
+            expectedsize = size
+
+        s = c * size
+        self.assertEquals(len(s.encode(enc)), expectedsize)
+
     @bigmemtest(minsize=_2G + 2, memuse=3)
     def test_encode(self, size):
-        s = '.' * size
-        self.assertEquals(len(s.encode('utf-8')), size)
+        return self.basic_encode_test(size, 'utf-8')
+
+    @precisionbigmemtest(size=_4G / 6 + 2, memuse=2)
+    def test_encode_raw_unicode_escape(self, size):
+        try:
+            return self.basic_encode_test(size, 'raw_unicode_escape')
+        except MemoryError:
+            pass # acceptable on 32-bit
+
+    @precisionbigmemtest(size=_4G / 5 + 70, memuse=3)
+    def test_encode_utf7(self, size):
+        try:
+            return self.basic_encode_test(size, 'utf7')
+        except MemoryError:
+            pass # acceptable on 32-bit
+
+    @precisionbigmemtest(size=_4G / 4 + 5, memuse=6)
+    def test_encode_utf32(self, size):
+        try:
+            return self.basic_encode_test(size, 'utf32', expectedsize=4*size+4)
+        except MemoryError:
+            pass # acceptable on 32-bit
+
+    @precisionbigmemtest(size=_2G-1, memuse=2)
+    def test_decodeascii(self, size):
+        return self.basic_encode_test(size, 'ascii', c='A')
+
+    @precisionbigmemtest(size=_4G / 5, memuse=6+2)
+    def test_unicode_repr_oflw(self, size):
+        try:
+            s = "\uAAAA"*size
+            r = repr(s)
+        except MemoryError:
+            pass # acceptable on 32-bit
+        else:
+            self.failUnless(s == eval(r))
 
     @bigmemtest(minsize=_2G, memuse=2)
     def test_endswith(self, size):
@@ -458,6 +515,11 @@
         self.assertEquals(s.count('\\'), size)
         self.assertEquals(s.count('0'), size * 2)
 
+    @bigmemtest(minsize=2**32 / 5, memuse=6+2)
+    def test_unicode_repr(self, size):
+        s = "\uAAAA" * size
+        self.failUnless(len(repr(s)) > size)
+
     # This test is meaningful even with size < 2G, as long as the
     # doubled string is > 2G (but it tests more if both are > 2G :)
     @bigmemtest(minsize=_1G + 2, memuse=3)
@@ -641,6 +703,35 @@
     def test_repeat_large(self, size):
         return self.basic_test_repeat(size)
 
+    @bigmemtest(minsize=_1G - 1, memuse=12)
+    def test_repeat_large_2(self, size):
+        return self.basic_test_repeat(size)
+
+    @precisionbigmemtest(size=_1G - 1, memuse=9)
+    def test_from_2G_generator(self, size):
+        try:
+            t = tuple(range(size))
+        except MemoryError:
+            pass # acceptable on 32-bit
+        else:
+            count = 0
+            for item in t:
+                self.assertEquals(item, count)
+                count += 1
+            self.assertEquals(count, size)
+
+    @precisionbigmemtest(size=_1G - 25, memuse=9)
+    def test_from_almost_2G_generator(self, size):
+        try:
+            t = tuple(range(size))
+            count = 0
+            for item in t:
+                self.assertEquals(item, count)
+                count += 1
+            self.assertEquals(count, size)
+        except MemoryError:
+            pass # acceptable, expected on 32-bit
+
     # Like test_concat, split in two.
     def basic_test_repr(self, size):
         t = (0,) * size