How can we use numpy's random `integers` function to get uniformly selected
integers from an arbitrarily large `high` limit? This is important when dealing
with exact probabilities in combinatorially large solution spaces.
I propose that we add the capability for `integers` to construct arrays of type
object_ by having it construct python int's as the objects in the returned
array. This would allow arbitrarily large integers.
The Python random library's `randrange` constructs values for arbitrary upper
limits -- and they are exact when using subclasses of `random.Random` with a
`getrandbits` methods (which includes the default rng for most operating
systems).
Numpy's random `integers` function rightfully raises on `integers(20**20,
dtype=int64)` because the upper limit is above what can be held in an `int64`.
But Python `int` objects store arbitrarily large integers. So I would expect
`integers(20**20, dtype=object)` to create random integers on the desired
range. Instead a TypeError is raised `Unsupported dtype dtype('O') for
integers`. It seems we could provide support for dtype('O') by constructing
Python `int` values and this would allow arbitrarily large ranges of integers.
The core of this functionality would be close to the seven lines used in [the
code of
random.Random._randbelow](https://github.com/python/cpython/blob/eb953d6e4484339067837020f77eecac61f8d4f8/Lib/random.py#L242)
which
1) finds the number of bits needed to describe the `high` argument.
2) generates that number of random bits.
3) converts them to a python int and checks if it is larger than the input
`high`. If so, repeat from step 2.
I realize that people can just use `random.randrange` to obtain this
functionality, but that doesn't return an array, and uses a different RNG
possibly requiring tracking two RNG states.
This text was also used to create [Issue
#24458](https://github.com/numpy/numpy/issues/24458)
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