Fixed issue in CRT_vectors where moduli are not allowed to be coprime.

[Noel-Roemmele]

Fixes #39158. Fixed the issue by creating functionality for CRT_basis if the moduli are not coprime. A more in depth explanation on how CRT_basis was expanded can be found in Here.

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

[Noel-Roemmele]

@DaveWitteMorris. I have completed the expansion for CRT_vectors and CRT_basis as we discussed. I do think it still needs amore detailed explanation in CRT_basis as to what an approximate CRT basis is versus a formal CRT basis. I also still need a good test for the new functionality in CRT_basis.

[Noel-Roemmele]

@user202729. Sorry to mention you but it seems like an additional bug detailed Here might also be resolved by this PR. I just wanted to ask if you can confirm that this bug is related and solved by this PR? If it is I will add a Fixes thing at the top.

[user202729]

It (probably?) only fix the second bug (moduli must be coprime), not the first, so I wouldn't count.

[github-actions]

Documentation preview for this PR (built with commit 4c17ea3; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[DaveWitteMorris]

The code in lines 3695-3701 is an exact copy of the code in lines 3686-3692. You should refactor to eliminate this code duplication. One possibility might be something like:

try:
    M = prod(moduli)
            ...
        if not d.is_one():
            raise ValueError('moduli must be coprime')
        cs.append((v * Mm) % M)
    if require_coprime_moduli:
        return cs
    # also return a boolean flag to report that the moduli are coprime
    return [cs, True]
except ValueError:
    if require_coprime_moduli:
        raise
    e = [1]
        ...

[Noel-Roemmele]

@DaveWitteMorris I have applied your suggestions and removed the code duplication.

[DaveWitteMorris]

Before the construction of the CRT basis for non-coprime moduli (starting at line 3708), I think you need to add a comment something like:

The following construction of a CRT basis for non-coprime moduli
is explained in a comment on :issue:`39158`.

[DaveWitteMorris]

Running doctests locally was successful, so I expect to set to positive review after the documentation typo is fixed.

[Noel-Roemmele]

@DaveWitteMorris. I've accepted the typo change. Thank you for pointing that out! Everything should be good for a positive review now.