diff --git a/Objects/longobject.c b/Objects/longobject.c
index 814ef642b48..9f414ca8a9d 100644
--- a/Objects/longobject.c
+++ b/Objects/longobject.c
@@ -1775,9 +1775,9 @@ _PyLong_Format(PyObject *aa, int base)
 	Py_ssize_t i, sz;
 	Py_ssize_t size_a;
 	Py_UNICODE *p;
-	int bits, negative;
+	int bits;
+	char sign = '\0';
 
-	assert(base == 2 || base == 8 || base == 10 || base == 16);
 	if (base == 10)
 		return long_to_decimal_string((PyObject *)a);
 
@@ -1785,34 +1785,23 @@ _PyLong_Format(PyObject *aa, int base)
 		PyErr_BadInternalCall();
 		return NULL;
 	}
+	assert(base >= 2 && base <= 36);
 	size_a = ABS(Py_SIZE(a));
-	negative = Py_SIZE(a) < 0;
 
 	/* Compute a rough upper bound for the length of the string */
-	switch (base) {
-	case 2:
-		bits = 1;
-		break;
-	case 8:
-		bits = 3;
-		break;
-	case 16:
-		bits = 4;
-		break;
-	default:
-		assert(0); /* never get here */
+	i = base;
+	bits = 0;
+	while (i > 1) {
+		++bits;
+		i >>= 1;
 	}
-
-	/* length of string required: +1 for negative sign, +2 for prefix */
 	i = 5;
 	/* ensure we don't get signed overflow in sz calculation */
-	if (size_a > (PY_SSIZE_T_MAX - 4) / PyLong_SHIFT) {
+	if (size_a > (PY_SSIZE_T_MAX - i) / PyLong_SHIFT) {
 		PyErr_SetString(PyExc_OverflowError,
 				"int is too large to format");
 		return NULL;
 	}
-	sz = 3 + negative + (size_a * PyLong_SHIFT - 1) / bits;
-
 	sz = i + 1 + (size_a * PyLong_SHIFT - 1) / bits;
 	assert(sz >= 0);
 	str = PyUnicode_FromUnicode(NULL, sz);
@@ -1820,41 +1809,112 @@ _PyLong_Format(PyObject *aa, int base)
 		return NULL;
 	p = PyUnicode_AS_UNICODE(str) + sz;
 	*p = '\0';
+	if (Py_SIZE(a) < 0)
+		sign = '-';
 
 	if (Py_SIZE(a) == 0) {
 		*--p = '0';
 	}
-	assert((base & (base-1)) = 0);
-	/* JRH: special case for power-of-2 bases */
-	twodigits accum = 0;
-	int accumbits = 0;	/* # of bits in accum */
-	int basebits = 1;	/* # of bits in base-1 */
-	i = base;
-	while ((i >>= 1) > 1)
-		++basebits;
+	else if ((base & (base - 1)) == 0) {
+		/* JRH: special case for power-of-2 bases */
+		twodigits accum = 0;
+		int accumbits = 0;	/* # of bits in accum */
+		int basebits = 1;	/* # of bits in base-1 */
+		i = base;
+		while ((i >>= 1) > 1)
+			++basebits;
 
-	for (i = 0; i < size_a; ++i) {
-		accum |= (twodigits)a->ob_digit[i] << accumbits;
-		accumbits += PyLong_SHIFT;
-		assert(accumbits >= basebits);
+		for (i = 0; i < size_a; ++i) {
+			accum |= (twodigits)a->ob_digit[i] << accumbits;
+			accumbits += PyLong_SHIFT;
+			assert(accumbits >= basebits);
+			do {
+				char cdigit = (char)(accum & (base - 1));
+				cdigit += (cdigit < 10) ? '0' : 'a'-10;
+				assert(p > PyUnicode_AS_UNICODE(str));
+				*--p = cdigit;
+				accumbits -= basebits;
+				accum >>= basebits;
+			} while (i < size_a-1 ? accumbits >= basebits :
+						accum > 0);
+		}
+	}
+	else {
+		/* Not 0, and base not a power of 2.  Divide repeatedly by
+		   base, but for speed use the highest power of base that
+		   fits in a digit. */
+		Py_ssize_t size = size_a;
+		digit *pin = a->ob_digit;
+		PyLongObject *scratch;
+		/* powbasw <- largest power of base that fits in a digit. */
+		digit powbase = base;  /* powbase == base ** power */
+		int power = 1;
+		for (;;) {
+			twodigits newpow = powbase * (twodigits)base;
+			if (newpow >> PyLong_SHIFT)
+				/* doesn't fit in a digit */
+				break;
+			powbase = (digit)newpow;
+			++power;
+		}
+
+		/* Get a scratch area for repeated division. */
+		scratch = _PyLong_New(size);
+		if (scratch == NULL) {
+			Py_DECREF(str);
+			return NULL;
+		}
+
+		/* Repeatedly divide by powbase. */
 		do {
-			char cdigit = (char)(accum & (base - 1));
-			cdigit += (cdigit < 10) ? '0' : 'a'-10;
-			assert(p > PyUnicode_AS_UNICODE(str));
-			*--p = cdigit;
-			accumbits -= basebits;
-			accum >>= basebits;
-		} while (i < size_a-1 ? accumbits >= basebits :
-			 accum > 0);
+			int ntostore = power;
+			digit rem = inplace_divrem1(scratch->ob_digit,
+						     pin, size, powbase);
+			pin = scratch->ob_digit; /* no need to use a again */
+			if (pin[size - 1] == 0)
+				--size;
+			SIGCHECK({
+				Py_DECREF(scratch);
+				Py_DECREF(str);
+				return NULL;
+			})
+
+			/* Break rem into digits. */
+			assert(ntostore > 0);
+			do {
+				digit nextrem = (digit)(rem / base);
+				char c = (char)(rem - nextrem * base);
+				assert(p > PyUnicode_AS_UNICODE(str));
+				c += (c < 10) ? '0' : 'a'-10;
+				*--p = c;
+				rem = nextrem;
+				--ntostore;
+				/* Termination is a bit delicate:  must not
+				   store leading zeroes, so must get out if
+				   remaining quotient and rem are both 0. */
+			} while (ntostore && (size || rem));
+		} while (size != 0);
+		Py_DECREF(scratch);
 	}
 
-	if (base == 16)
+	if (base == 16) {
 		*--p = 'x';
-	else if (base == 8)
+		*--p = '0';
+	}
+	else if (base == 8) {
 		*--p = 'o';
-	else /* base == 2 */
+		*--p = '0';
+	}
+	else if (base == 2) {
 		*--p = 'b';
-	*--p = '0';
+		*--p = '0';
+	}
+	else if (base != 10) {
+		*--p = '#';
+		*--p = '0' + base%10;
+		if (base > 10)
+			*--p = '0' + base/10;
+	}
 	if (sign)
 		*--p = sign;
 	if (p != PyUnicode_AS_UNICODE(str)) {