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------------ current state ------------
On my Windows 10, on a 32-bit Python, The following runs fine:
 import array
 array.array('L').append(2 ** 32 - 1)
However, in the following, an OverflowError('Python int too large to convert to C long') is raised on the last line:
 import array
 class LikeInt:
 def __init__(self, intVal):
 self.intVal = intVal
 def __int__(self):
 return self.intVal
 array.array('L').append(LikeInt(2 ** 32 - 1))
The reason for this behavior is the implementation of the function LL_setitem (in Modules/arraymodule.c) (edited brutally for brevity):
 LL_setitem(arrayobject *ap, Py_ssize_t i, PyObject *v)
 {
 unsigned long x;
 if (PyLong_Check(v)) {
 x = PyLong_AsUnsignedLong(v);
 }
 else {
 long y;
 PyArg_Parse(v, "l;array item must be integer", &y);
 x = (unsigned long)y;
 }
 (ap->ob_item)[i] = x;
 }
The problem is that PyArg_Parse is used to convert a Python int into a C long. So PyArg_Parse fails when it is given a Python int which is in range(LONG_MAX + 1, ULONG_MAX + 1), even though such Python int can be stored in a C unsigned long.
It is quite the same for array('I') and array('Q') (i.e. in II_setitem and in QQ_setitem, respectively).
With regard to relevant changes made in the past, PyArg_Parse was always used (in case '!PyLong_Check(v)'), since adding the original versions of:
 * II_setitem and LL_setitem (back then, they were called I_setitem and L_setitem, respectively), in changeset 4875 (https://hg.python.org/cpython/rev/911040e1bb11)
 * QQ_setitem, in changeset 72430 (https://hg.python.org/cpython/rev/15659e0e2b2e)
------------ proposed changes ------------
 1. In Modules/arraymodule.c, change the implementation of LL_setitem (and likewise, of II_setitem and QQ_setitem) roughly to the following:
 LL_setitem(arrayobject *ap, Py_ssize_t i, PyObject *v)
 {
 unsigned long x;
 if (!PyLong_Check(v)) {
 v = _PyLong_FromNbInt(v);
 }
 x = PyLong_AsUnsignedLong(v);
 (ap->ob_item)[i] = x;
 }
 2. In Lib/test/test_array.py, add tests:
 * to verify the bug is fixed, i.e. test the bounds of setting int-like objects to items in an integers array
 * to verify float objects can't be set to items in an integers array (as is already the behavior, but there aren't any tests for it)
 3. While we are in Lib/test/test_array.py, remove any checks whether 'long long' is available, as it must be available, since changeset 103105 (https://hg.python.org/cpython/rev/9206a86f7321).
Note that issue #12974 (opened in 2011) proposes deprecating the ability to set int-like objects to items in an integers array. (I am not sure how that affects my patch, but I guess it should be noted.)
------------ diff ------------
The proposed patches diff file is attached.
(Note that the I didn't propose to change the perplexing error message 'unsigned int is greater than maximum' in II_setitem, as there are many such messages in the codebase, and I am working on a patch for them as part of issue #15988.)
------------ tests ------------
I ran 'python_d.exe -m test -j3' (on my 64-bit Windows 10) with and without the patches, and got quite the same output. (That also means my new tests in test_array passed.)
The outputs of both runs are attached.

One comment here: it's not the presence of `__int__` that makes a type integer-like; it's the presence of `__index__`. (Decimal and float both supply `__int__`, but shouldn't be regarded as integer-like, for example.) I'm guessing that we're going to have the same issues with `__index__` in place of `__int__`, though.

You are right about the terminology of course. My bad.
Anyway, the bug is not related to __index__ (or nb_index), because the three aforementioned functions are using (in case '!PyLong_Check(v)') PyArg_Parse with the formats 'l' or 'L'.
Ultimately, convertsimple (in Python/getargs.c) would call PyLong_AsLong or PyLong_AsLongLong (respectively).
These two (in case '!PyLong_Check(v)') only try to use nb_int, as documented:
 * https://docs.python.org/3.7/c-api/long.html?highlight=pylong_aslong#c.PyLong_AsLong
 * https://docs.python.org/3.7/c-api/long.html?highlight=pylong_aslong#c.PyLong_AsLongLong
But just in case, I ran the following:
 import array
 class LikeInt:
 def __init__(self, intVal):
 self.intVal = intVal
 def __index__(self):
 return self.intVal
 array.array('L').append(LikeInt(0))
and got a 'TypeError: an integer is required (got type LikeInt)'

> and got a 'TypeError: an integer is required (got type LikeInt)'
Right, see #12974 for more on arrays, `__int__` and `__index__`.

Ah, I should have read more about __int__ and __index__ before writing my last reply.
So IIUC, this is a somewhat painful issue, which could be resolved by:
 * #20092
 * replacing _PyLong_FromNbInt with PyNumber_Index in:
 - Objects/longobject.c (as you suggested in https://bugs.python.org/issue12965#msg146252)
 - Modules/zlibmodule.c in ssize_t_converter
I didn't find any issue for the second one. If there is really no such issue, why is that? (I am sorry if I am the thousandth one to ask)
The first paragraph of your comment in https://bugs.python.org/issue21111#msg215660 suggests it might break a lot of code in the wild, but shouldn't we start with raising a deprecation warning?
With regard to my patch - should I make the following changes?
 1. in Modules/arraymodule.c - add a call to PyNumber_Index before the call to _PyLong_FromNbInt (and move them into a helper function, as done in Modules/_struct.c)
 2. in Lib/test/test_array.py:
 * replace the name 'LikeInt' with 'IntCompatible' (I am not sure this is an appropriate name either. If you have a better name in mind, please share :) )
 * add tests for LikeInt (this time, a class with a __index__ method)
 3. In Doc/library/array.rst:
 * add a note saying that setting int-like objects (i.e. objects with __index__) to items in an integers array is possible
 * while we are here, remove the note about the availability of 'q' and 'Q' only in certain platforms

ping

LGTM. But maybe combine PyFloat_Check+_PyLong_FromNbInt in one helper function?
Could you please create a PR Oren?

yes and yes.
I would start with a PR for 3.7.
and thanks for the review :)

Thank you for your contribution Oren.

Thanks for the reviews :)

New changeset e2c88bdd6bb3efbc81389958d62daf6dd0d6eda7 by Serhiy Storchaka (orenmn) in branch '3.5':
bpo-28298: make array 'Q', 'L' and 'I' accept big intables as elements
https://github.com/python/cpython/commit/e2c88bdd6bb3efbc81389958d62daf6dd0d6eda7

New changeset 26d013e00f9d4adbcf3e084bbc890c799ff70407 by Serhiy Storchaka (orenmn) in branch '3.6':
[3.6] bpo-28298: make array 'Q', 'L' and 'I' accept big intables as elements (#579)
https://github.com/python/cpython/commit/26d013e00f9d4adbcf3e084bbc890c799ff70407

New changeset 964281af59d7a17d923c4d72357e48832b774e39 by Serhiy Storchaka (orenmn) in branch 'master':
bpo-28298: make array 'Q', 'L' and 'I' accept big intables as elements (#570)
https://github.com/python/cpython/commit/964281af59d7a17d923c4d72357e48832b774e39