This website provides scripts and log files related to the proof of the 451-Theorem. (For the paper, see this file.) The table below gives a listing of the files, file types, and short descriptions.
Note: At the end of May 2016, I discovered a possible bug in the Magma script prep6.txt that computes the cusp constant. This affects the method 4 computations only. I fixed the bug and reran all the method 4 computations. There were no differences in the cusp constants that were computed originally with those that were recomputed. For this reason, I updated the file prep6.txt, but I have left only the original log files.
| Filename | File type | Description |
| escalateodd.txt | Magma script | This script computes 1-dimensional, 2-dimensional, 3-dimensional and 4-dimensional escalations of the zero dimensional lattice and writes the Gram matrices to a file. The file of 4-dimensional escalators is e4.txt, which is used by other programs. |
| univ3.txt | Magma list | This is a list of the Gram matrices of the 23 ternary quadratic forms given by Kaplansky. It is output by escalateodd.txt, and is loaded by prep6.txt and univquat5.txt. |
| 1115switch.txt | Magma script | This script computes escalations of x2 + 3y2 + 5z2 by 15. This is one type of auxiliary escalator. |
| 1115out.txt | Magma list | This is the output of 1115switch.txt. |
| 1115reduce.txt | Magma script | This script removes duplicate lattices from e4.txt and 1115out.txt. The output is written to 1115red.txt. |
| 1115red.txt | Magma list | This is the output of 1115reduce.txt. |
| 1321switch.txt | Magma script | This script computes escalations of x2 + xy + 3y2 + 7z2 by 21. This is the other type of auxiliary escalator. |
| 1321out.txt | Magma list | This is the output of 1321switch.txt. |
| 1321reduce.txt | Magma script | This script checks to see which lattices occur in 1321out.txt also occur in e4.txt or 1115red.txt. Any that do not are written to 1321red.txt. |
| 1321red.txt | Magma list | This is the output of 1321reduce.txt. |
| quatver.txt | Magma list | This is the list of all 24888 quaternary escalators. It is the result of removing the four forms that do not locally represent all odd numbers from e4.txt, and then concatenating with 1115red.txt and 1321red.txt. A form referred to by number indicates its index in this list. This list is used by most Magma scripts listed later, including prep6.txt. |
| beslistmake.txt | Magma script | This script pre-computes values of the |
| goodlats2.out | Magma list | This is a list indicating which of the 24888 quaternary escalators has a nicely embedded regular ternary sublattice. It is called by prep6.txt. |
| reglats.txt | Magma list | This is a list of the 891 provably-regular ternary quadratic forms as enumerated by Jagy, Kaplansky, and Schiemann. It is used by prep6.txt. |
| method1.out | Magma list | This is a list of the indices of the 2342 forms for which method 1 (a universal ternary sublattice) is used. |
| method2.out | Magma list | This is a list of the indices of the 7470 forms for which method 2 (a nicely embedded regular ternary) is used. |
| method3.out | Magma list | This is a list of the indices of the 8733 forms for which
method 3 (Rankin-Selberg |
| method4.out | Magma list | This is a list of the indices of the 6343 forms for which method 4 (explicit computation of the cusp constant) is used. |
| univternlist.txt | Magma list | This is a list of which universal ternary is represented by each of the 2342 method 1 forms. |
| prep6.txt | Magma script | This script does the bulk of the work in proving the 451-Theorem. It takes as a command line argument an index in the list quatver.txt and does all of the theoretical work in determining the squarefree integers that quadratic form represents. Appropriate command line syntax would be something like "magma mmm:=16366 prep6.txt". If this script is run on form N, it creates two output files: N.log and Ndata.out. The second of these is fed into the C program represent2.c, if necessary. This script loads the files quatver.txt, goodlats2.out, reglats.txt, method1.out, method2.out, method3.out, method4.out, univternlist.txt, univ3.txt, and beslist.out. |
| primelist.gp | GP script | This is a script written in PARI/GP that computes the odd primes less than 300000000 (all 16252324 of them) and writes them to the file primelist.txt. This file is read by represent2.c. |
| represent2.c | C source code | This is the source code for a program that checks integers less than the F4 bound given in the output file by prep6.txt. It takes as a command line argument the output file, and writes more data to the .log file. |
| represent2big.c | C source code | This is very similar to the source code above. The only essential difference is the constant on line 689 is 64 instead of 16. This is necessary for testing forms with large anisotropic primes. |
| method1logs.tgz | tar-gzip file | This file contains the output and log files for the 2342 quaternaries where method 1 was used. |
| method2logs.tgz | tar-gzip file | This file contains the output and log files for the 7470 quaternaries where method 2 was used. |
| method3logs.tgz | tar-gzip file | This file contains the output and log files for the 8733 quaternaries where method 3 was used. |
| method4logs.tgz | tar-gzip file | This file contains the output and log files for the 6343 quaternaries where method 4 was used. |
| looseend.txt | Magma script | This file is a script for checking 5-dimensional escalators to see if they fail to represent any integer on a fixed list. It is used to complete the proof of the 451-Theorem. |
| looseend.16366 | Text file | This file gives the output of the script looseend.txt run on form 16366. This proves that no escalation of 16366 represents 627, 935, 1111, 1419, 1903, or 2387. |
| isembeddable.txt | Magma script | This file is a script containing a single function that
takes as input a ternary lattice |
| univquat5.txt | Magma script | This file enumerates all quaternary quadratic forms that represent all odd integers but do not represent any of the odd universal ternaries. It uses the 451-Theorem, and outputs the result as univquats3.txt. The running time required is about 11 hours. This script calls isembeddable.txt. |
| univquats3.txt | Magma list | This file lists all 21756 quaternary quadratic forms that represent all odd integers and do not represent any of the odd universal ternaries. It is the output of univquat5.txt. |