Because I haven't added a new page to my site for a long time, it is going to take me a while to remember what to do... . Also, my site went down after an attack... TESTING TESTING TESTING
A note imported from the Maple section of my website.
You may encounter a problem opening my Maple worksheets (as indeed I do myself) here at my web site - it would appear to depend on the internet
browser being used.
Thus, if I attempt to open one of my worksheets using Interner Explorer there is never a problem, whereas if I use Firefox (or Google Chrome) then all
that one sees - this is just an example - is something like this:
{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 0 128 0 128 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 128 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE ""
and so on (almost at infinitum)...
I asked Maplesoft for advice on this and they recommended doing the following (and I found it worked):
- Don't attempt directly to open the worksheet (by clicking on my link), instead right click and save the worksheet to (say) the download folder.
- Now that the worksheet is in the download folder (cut it out if you wish and put into whatever folder you wish) you may open the worksheet
in the usual way (I should add that while I see this 'works', I have absolutely no idea as to why it does... ) [END OF NOTE].
These are the headings of the 18 sections in my Maple-Wieferich document:
- Section 1. Introduction (Mer(p) - the full Mersenne group - and Aux(p), the auxiliary Mersenne group)
- Section 2. A dilemma (how much should I explain...), and a selection of Wieferich primes references
- Section 3. Two results concerning the square-free nature of Mersenne Mp = 2p - 1 and Fermat Fn = 22n + 1, are seen to be particular cases of a much wider phenomenon.
And a new (?) square-free observation, one which - if true - has consequences
- Section 4. The start of my story (some background to my 1999 conjectured primality test for Mersenne primes)
- Section 5. How I was led, in April 1999, to formulate the proposed primality test of Section 3
- Section 6. The first step in the right direction: the determination of Mer(11).
- and - taken out of sequence - the structure theorem for the general Mersenne group Mer(p).
Theorem 6.1 (that Mer(p) is a multiplicative group).
A numerical illustration of a critical difference between the Wieferich prime p = 1093 and non-Wieferich primes with respect to
the Mersenne group Mer(p) (I don't suppose that virtually all primes would like to be referred to as being non-Wieferich primes!)
- Section 7. Theorems 7.1 and 7.2: the p-subgroup (singular) of Mer(p) for non-Wieferich primes.
Theorem 7.3. The structure of Mer(p) for non-Wieferich primes.
- Section 8. Theorem 8.1: the p-subgroups (plural) of Mer(p) for Wieferich primes.
Theorem 8.2. The structure of Mer(p) for Wieferich primes.
- Section 9. Some more progress: the determination (87 minutes) of Mer(23), and a verification of its group structure.
- Section 10. Some hard-fought progress: the determination (4 days in total, with freezes discounted) of Mer(29).
I have included a verification of its group structure.
- Section 11. The determination of Mer(29) seemed to be the end of the journey until: my fortuitous, critical development.
The fundamental object:
gcd( φ(Mp)/p, (Mp - 1)/p ),
and now any Aux(p) may be determined in minutes, providing one knows the complete prime factorization of Mp.
- Section 12. Statement of the Main Theorem (I lack the html skills necessary to render its statement here; readers will have to see it at the downloaded Maple page).
- Section 13. Two standard, well known results concerning the number of solutions of simple power congruences, with some illustrative examples, including use of the Chinese Remainder Theorem [and something about Cai Tianxin, the contemporary Chinese mathematician/poet]
- Section 14. The construction of the 10935 elements of the 1093-subgroup of Mer(1093).
- Section 15. A hymn to the Euclidean algorithm (for grandson Feidhlim)
- Section 16. Some reflections and new conjectures/questions concerning the { Mer(p) }, p prime.
This section contains 4 sub-sections:
16.1. A new (?) question concerning the classic (remarkable) Lucas-Lehmer primality test.
16.2. Are there infinitely many composite Mer(p)?
16.3. Determining complete factorizations of composite Mer(p).
16.4. The classic conjecture that the { Mer(p) } are square-free, and a new (?) square-free question.
- Section 17. Consequences of (6.i), with something left for readers to investigate.
- Section 18. Addendum on rare primes (and introducing Lagrange primes), and an Iwasawa Theory connection.
This section contains 8 sub-sections:
18.1. First, a (very hard) problem for readers.
18.2. The prime p = 5 (and more besides).
18.3. The prime p = 53 (and more besides).
18.4. The prime p = 59 (and more besides).
18.5. A (new?) Lagrange question (a research problem for anyone who is interested).
18.6. The prime p = 76543 (Iwasawa Theory connection (made by someone else, to be announced soon)).
18.7. The prime p = 29789 (again cutting corners) (Iwasawa Theory connection made by someone else, to be announced soon).
18.8. The known Euler primes are 149, 241, 2946901, 16467631, 17613227, 327784727, 426369739, 1062232319.
