Computations
This page is essentially complete, though further computations may be added from time to time.
A reader of this page who does not have (the wonderful software) Maple should download the FREE 'Maple Player' from the Maplesoft site. This player - as explained below - enables anyone to read (but not alter, or experiment with) the contents of a Maple worksheet.
On July 13th 2015 Karl Dilcher and I submitted our tenth paper - A role for generalized Fermat numbers - to the American Mathematical Society journal Mathematics of Computation. Our paper was formally accepted on September 18th 2015, and it has now been published:
| Journal | Math. Comp. 86 (2017), 899-933 |
|---|---|
| Authors | John B. Cosgrave and Karl Dilcher |
| MSC (2010) | Primary 11A07; Secondary 11B65 |
| DOI: | https://doi.org/10.1090/mcom/3111 |
| Published Electronically | April 26, 2016 |
| MathSciNet Review | 3584554 |
Our earlier papers are listed, with some commentary, in the publications section of this site.
This tenth paper is the natural successor of our earlier:
THE GAUSS–WILSON THEOREM FOR QUARTER-INTERVALS, Acta Mathematica Hungarica, 142 (1) (2014), 199–230
There are well over one hundred supporting Maple worksheets accompanying 'A role ... '. All of these worksheets are available at this site, and others will be added in the future. The ideal reader is someone who has Maple, but these worksheets may be read by anyone (even those who do not have Maple), as I explain below.
First, though, some remarks aimed at Maple users. A reader of our paper who uses Maple may download any of these worksheets and verify the claimed outputs, or may wish to write their own programs or procedures to independently verify.
Read-only file. This is a read only file. You may not be able to save any changes.
A suggested solution for a reader who wishes to carry out their own independent work with an opened worksheet: having opened a worksheet, click on the 'OK' box to enter the worksheet. Next got to 'Edit', then 'Select All', and copy and paste into a newly opened Maple worksheet; finally save with a new name.
Next, some remarks for those who do not have Maple. (I am grateful to Dr. Robert Lopez of Maplesoft for bringing the following to my attention): every Maple worksheet below may be read (but not edited) by saving it, and then opening it with the free Maple Reader package.
Before I knew this could be done I had converted a limited number of the original Maple worksheets to pdf format (to get those done in a visually satisfactory manner was quite time consuming, and was a disincentive to doing so for all the others). Nevertheless, I have decided not to remove the pdf conversions, but I repeat: every (active) Maple worksheet below may be read using either Maple itself, or using the free 'Maple Reader'.
I hope I may be forgiven for spelling out in complete detail HOW any of the following may be read with Maple Reader (I can all too easily remember my own frustrations when first using software: how do you do this, how do you do that?). Suppose you wish to view one of the twenty zipped worksheets at #6 (the first of the non-pdf converted documents), then proceed as follows:
- (a) First download the zipped folder (its name is "Gauss 3 solutions for 1st 20 standard"). In that folder you will see 20 Maple worksheets, with names ranging from "G3 7 (1st solution at s = 7).mws" through to "G3 92401.mws"
- (b) Then, for example, to view the contents of G3 92401.mws, simply open Maple Player, go to File (in top left-hand corner), click on Open, and locate the downloaded G3 92401.mws.
What you see looks like a regular Maple worksheet (one which you won't be able to modify). The triangular regions are Maple 'Sections', any one of which may be opened by clicking on the triangle; you may close a section by once again clicking on the triangle.
#1. We begin with pdf converted Tables showing all solutions to \(10^{6}\) of congruences (2.5) and (2.6) of our paper.
Early Solutions 2.5 and 2.6 (PDF, 399.48 KB)
#2. Here is a pdf Table of all 20 standard Jacobi primes to \(10^{5}\).
Level 0 Jacobi's 10^5 (PDF, 161.61 KB)
#3. Here is a pdf Table of all 45 nonstandard Jacobi primes to \(10^{14}\)
Non-standard Jacobi's 10^14 (PDF, 325.34 KB)
#4. Next are 12 zipped Maple worksheets (in pdf format) showing how we determined all nonstandard Jacobi primes (those of levels 3 through to 19; there are 33 of them).
45 Non-standard PDFs (ZIP, 3.78 MB)
#5. And here is a single Maple worksheet (in pdf format) showing how we determined all nonstandard Jacobi primes (those of level at least 20; there are 12 of them).
Jacobi's to 10^14 of Level at Least 20 (PDF, 159.34 KB)
#6. Here are 20 zipped Maple worksheets treating (for the case 's' > 1) the first 20 standard Jacobi primes for the \(M = 3\) case of our paper.
Gauss 3 Solutions for 1st 20 Standard (ZIP, 405.11 KB)
#7. Here are 45 zipped Maple worksheets treating (for the case 's' > 1) the 45 nonstandard Jacobi primes to \(10^{14}\) for the \(M = 3\) case of our paper.
Gauss 3 Solutions for 45 Non-standard (ZIP, 566.19 KB)
#8. Here are 20 zipped Maple worksheets treating (for the case 's' > 1) the first 20 standard Jacobi primes for the \(M = 6\) case of our paper.
Gauss 6 Solutions for 1st 20 Standard (ZIP, 286.48 KB)
#9. Here are 45 zipped Maple worksheets treating (for the case 's' > 1) the 45 nonstandard Jacobi primes to \(10^{14}\) for the \(M = 6\) case of our paper.
Gauss 6 Solutions for 45 Non-standard (ZIP, 635.88 KB)
#10. Here are 4 zipped Maple worksheets (in pdf format) treating Examples 2.2 and 2.3 of our paper.
#11. Here are 4 zipped Maple worksheets (in pdf format) treating Examples 7.8 and 7.11 of our paper.
Examples 2.2 and 2.3 Expanded (ZIP, 3.07 MB)
Examples 7.8 and 7.11 Expanded (ZIP, 3.62 MB)
#12. Here is a pdf Table showing complete factoring (to as high a level as currently possible) for the generalised Fermat numbers \(\frac{1}{2}\left(p^{2^{j}} + 1\right)\) for the (twenty) standard Jacobi primes to \(10^{5}\).
Factor Tables 11 (PDF, 104.82 KB)
#13. Here is a pdf Table showing complete factoring (to as high a level as currently possible) for the generalised Fermat numbers \(\frac{1}{2}\left(p^{2^{j}} + 1\right)\) for the (forty-five) nonstandard Jacobi primes to \(10^{14}\).
Factor Tables 12 (PDF, 114.33 KB)