BQFRepCheck

Installation instructions: see below.

This repository contains code to computationally check which integers can be represented by a shifted binary quadratic form, comparing it to the theoretical statement in Theorems 2 and 3 of "Primes of the type ϕ(x,y)+A where ϕ is a quadratic form", by Iwaniec, which can be found here.

Ensuring the correctness of the tables and correcting the minor errors was critical for the proofs in the paper Primes represented by shifted quadratic forms: on primitivity and congruence classes, by Fuchs, Hsu, Rickards, Schindler, and Stange.

Citation

Check the "Cite this repository" on the side bar (under "About") for a sample BibLaTeX entry.

Installation Instructions

Downloading the code

Call git clone https://github.com/JamesRickards-Canada/BQFRepCheck.git. If you are on Windows, be sure to git clone from WSL (see below), as Windows line endings (carriage returns) may be added to files, causing issues.

Prerequisites

• PARI/GP, but not the downloaded ready-to-go binary. The PARI/GP website has binaries for Windows and Mac avaliable, but these will not work with the package. See below for OS specific instructions.
• This was tested on version 2.18.0, but should work with any version released since 2019.
• You should have a guess as to the location of the pari.cfg file for the version of PARI/GP you are running. Suggestions on how to do this can be found below.

Operating systems

• Linux - No further requirements
• Windows - You need to use Windows Subsytem for Linux. See the guide I wrote for additional instructions.
• Mac - You need to have Homebrew installed. This is also an easy way to install PARI/GP: brew install pari

Where is pari.cfg?

• The configuration file will search for this, but it is preferrable to not search your entire hard drive (as this can be very slow). So, you should at least supply a guess as to the location of pari.cfg. Often only the top-level folder (e.g. /usr or /opt) suffices.
• On Linux or WSL, if you build PARI/GP from source, it should be located in /usr/local/lib/pari/pari.cfg, or at least somewhere in the /usr folder.
• On a Mac, if you install PARI/GP with Homebrew, it may be found in a folder like /opt/homebrew/Cellar/pari/VERSION/lib/pari. Searching /opt/homebrew should be fine.
• If you are obtaining it through SageMath, it might be found where the library files of SageMath are
• Assuming you open PARI/GP with the command gp, try type -a gp, which will display where this command lives. The corresponding file(s) are likely symbolic links, and you can call readlink -f LOCATION on each of them to see where it lives. This can provide a clue as to the place to search for pari.cfg.
• Another clue comes from gp itself. Open gp, and type default(). Look for the entries datadir and help. It is often the case that datadir is in X/share/pari, help is in X/bin/gphelp, and pari.cfg is in X/lib/pari/pari.cfg.

Configuring and building the package

• From inside the project folder, call ./configure to initialize the project. This helps you search for pari.cfg, and stores the location to a file. You should supply it with a folder to search in!
• The script displays the corresponding versions of the found files, so if you have multiple versions, you can choose the correct one. This can be useful if you keep multiple copies of PARI/GP around.
• If the location of the installation of PARI/GP does not change, you do not need to reconfigure. If when you update PARI/GP there is a new location (e.g. if the version number is in the file path of pari.cfg), you should call ./configure again.
• Call make to build the project, and make clean to remove all .o object files. If you update to a new version of PARI/GP, you must remake the project.

Troubleshooting

• If you are encoutering unexpected errors or warnings when building, ensure you found the correct pari.cfg file. You need to configure the project with the same version of PARI/GP you are using.
• If you upgrade your version of PARI/GP, you should make clean, call ./configure again, and remake the project.
• If you still have trouble installing or using the package, please get in touch!

Using the code

Start gp and load the files with gp quadratic bqfcheck. The tests we claim in our paper can be accomplished with

dl = disclist(-1000, 1000);
for (i = 1, #dl, test_thm2(dl[i], -100000, 100000));

Any contradictions to the tables will produce an output. If the method does not print anything and finishes, all tests were passed.