diff -r 5683ca62a954 Lib/test/test_random.py
--- a/Lib/test/test_random.py	Tue May 08 21:29:36 2012 -0400
+++ b/Lib/test/test_random.py	Wed May 09 11:15:58 2012 +0100
@@ -57,6 +57,14 @@
             self.assertRaises(TypeError, self.gen.jumpahead)  # needs an arg
             self.assertRaises(TypeError, self.gen.jumpahead, 2, 3)  # too many
 
+    def test_jumpahead_produces_valid_state(self):
+        # From http://bugs.python.org/issue14591.
+        self.gen.seed(199210368)
+        self.gen.jumpahead(13550674232554645900)
+        for i in range(500):
+            val = self.gen.random()
+            self.assertLess(val, 1.0)
+
     def test_sample(self):
         # For the entire allowable range of 0 <= k <= N, validate that
         # the sample is of the correct length and contains only unique items
diff -r 5683ca62a954 Modules/_randommodule.c
--- a/Modules/_randommodule.c	Tue May 08 21:29:36 2012 -0400
+++ b/Modules/_randommodule.c	Wed May 09 11:15:58 2012 +0100
@@ -427,8 +427,19 @@
         mt[j] = tmp;
     }
 
-    for (i = 0; i < N; i++)
+    for (i = 0; i < N; i++) {
         mt[i] += i+1;
+        mt[i] &= 0xffffffffUL; /* for WORDSIZE > 32 machines */
+    }
+
+    /* Make sure the state is nonzero: in the unlikely event that mt[1] through
+       mt[N-1] are all zero, we set the MSB of mt[0]. */
+    tmp = 0;
+    for (i = 1; i < N; i++) {
+        tmp |= mt[i];
+    }
+    if (tmp == 0)
+        mt[0] = 0x80000000UL;
 
     self->index = N;
     Py_INCREF(Py_None);