# [book] History Of Theory Of Numbers.

wendy.krieger
wendy.krieger
This is the latest inwards book, a three volume histoy of theory of numbers.

Vol 1 has a whole chapter (IV) on sevenites, under the title of the form of Euler's Quotient, p 105-112. Weiferich is mention in passing, but the original question was raised by Abel in 1828, and answered by examples by Jacobi (1839), the decimal sevenites 3 and 487 at least by 1852 (Desmarest).

Neither Dickson nor David Wells (1975) seemed to have a general name.

It's prettty encrusted in mathematica runes, so this is not an easy read, and i suppose it's down to looking at the pretty pictures.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Now what you called "sevenites" are called "base b Wieferich primes" or "generalized Wieferich primes" *today* by a significant contingent of international mathematicians, texts notwithstanding. I have Oystein Ore's history of the theory of numbers wherein he talks of "aliquot parts" meaning proper divisors. This is old diction that few use today; I read other mathematicians in later works fondly recalling Ore's "quaint writing" with terms like "aliquot part" that aren't used today.

What was the title of the chapter? It wasn't "Sevenites", was it?

Dan had a concept we called "abstract prime factorization" that a significant contingent of international mathematicians call "prime signature." Now we call it "prime signature" too, because that enables others to better understand what we're talking about.

I didn't mind using "sevenite" when there didn't seem to be a canonical term for such a thing, though it never sat well. The Russians have a word "vokzal" meaning train station. It was a long time that I came to understand that this word is taken from the name of an actual train station - Vaux Hall. What's so special about Vaux Hall (I'm sure it's pretty). My Greek friend is of course bent against Turks. He says that the conquerors of Constantinople got there by asking "where's the city," and kept getting the reply ending in "to the city" thus that is how he says Istanbul got it's name (I don't think it's true. I think it's K(i)n(stan)tino(bul)is = Istanbul. Happens all the time. Look what they did to Cordoba <= Al Qartuba <= Kar-Juba. When we take over a foreign city we can't seem to pronounce it right). So in his mind Constantinople was renamed "To the city". The Italian word in common use for a bus is pullman, from an actual make. Sevenite is a similar term. I'd rather not have a general term from a specific instance (apparently 7 in base 18.)

I don't like "generalized Wieferich prime" because it is ten years long and I am not sure how to say "Wieferich" (then again my surname is a doozie, so you won't see any eponymous functions or discoveries from me, thank you). But it is what that is called and I think we should use it if we want to be understood.

Mathematica function for finding base-b Wieferich primes:
f[b_, lim_] := Select[Prime@ Range@ lim, Divisible[b^(# - 1) - 1, #^2] &]

Implementation for base 18:
f[18, 18^3]

Output:
{5, 7, 37, 331, 33923}

By this, it might've been called "fivites" just as easily ; ).

Dozens Disciple
Dan
Dozens Disciple
Joined: Aug 8 2005, 02:45 PM
icarus @ Oct 14 2015, 07:58 AM wrote: I don't like "generalized Wieferich prime" because it is ten years long and I am not sure how to say "Wieferich"
I assume it's German and pronounced /vi:fÉ™rÉªÃ§/.

If "generalized Wieferich prime" is too long to say, we could use the acronym GWiP.

Dozens Demigod
Double sharp
Dozens Demigod
Joined: Sep 19 2015, 11:02 AM
Dan @ Oct 14 2015, 01:20 PM wrote:
icarus @ Oct 14 2015, 07:58 AM wrote: I don't like "generalized Wieferich prime" because it is ten years long and I am not sure how to say "Wieferich"
I assume it's German and pronounced /vi:fÉ™rÉªÃ§/.

If "generalized Wieferich prime" is too long to say, we could use the acronym GWiP.
Yes, Arthur Wieferich (1884-1954) was a German mathematician and teacher from MÃ¼nster.

I like "GWiP": it's compact, sounds cute, and can readily be unpacked.

@Wendy: Thank you for the reference!

Dozens Disciple
Treisaran
Dozens Disciple
Joined: Feb 14 2012, 01:00 PM
icarus wrote:The Russians have a word "vokzal" meaning train station. It was a long time that I came to understand that this word is taken from the name of an actual train station - Vaux Hall. What's so special about Vaux Hall (I'm sure it's pretty).
Maybe it was the first they ever sampled, like the way the Slavs gave the name nemetski 'mute' to the first non-Slavs they encountered, the Germans. Or like the way Latin letters are called 'English letters' by my fellow countrymen, or the whole turn of events where the Arabs call their positional decimal digits 'Indian numerals' (as opposed to the sign-value Arabic-letter abjadi numerals), while the West calls their borrowed and changed forms of those numerals 'Arabic numerals' (to distinguish from Roman or Greek numerals).
icarus wrote:He says that the conquerors of Constantinople got there by asking "where's the city," and kept getting the reply ending in "to the city" thus that is how he says Istanbul got it's name (I don't think it's true. I think it's K(i)n(stan)tino(bul)is = Istanbul.
For much of the Ottoman period, the name was actually Islambul. The Jews of the time, however, shortened the name to Koshta, similarly to how Aristotle appears as Aristo in Jewish texts (but not in Arabic - there he is Aristatalis, appearing together with the astronomer Batlimus (Talmai in Hebrew) and the doctor Abuqrat). In contrast to all this, the Norse warriors reverently named Constantinople MiklagarÃ°r 'Great Yard'. It seems even the trolls in the hall of the mountain king thought the builders and inhabitants of such a splendid city might be giants.

wendy.krieger
wendy.krieger
Vauxhall is the first station out of London Victoria. One of the Russian Tsars, I believe Nikolas, was staying there. The line runs past Clapham Junction, and Battersea. That's Battersea power house you see on the Pink Floyd LP, with the pig flying over it.

It is supposed that the Russian foot of 304.8 mms comes at that time too, as well as using water at 62 F (16 2/3C) to define the weight-volume relationship.

The first volume of Dickson is a Dover reprint, and it shoud not cost a lot if you are interested. It should cost about 20&#036;, plus postage.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
Treisaran @ Oct 14 2015, 04:07 PM wrote:
icarus wrote:The Russians have a word "vokzal" meaning train station. It was a long time that I came to understand that this word is taken from the name of an actual train station - Vaux Hall. What's so special about Vaux Hall (I'm sure it's pretty).
Maybe it was the first they ever sampled, like the way the Slavs gave the name nemetski 'mute' to the first non-Slavs they encountered, the Germans.
Or like the way the Romans called all the Hellenes (ÎˆÎ»Î»Î·Î½ÎµÏ‚) by the name Graeci "Greeks", just because the first ones they encountered were an obscure Adriatic-island tribe called the Graikoi (Î“ÏÎ±Î¯ÎºÎ¿Î¹).

Or how Greeks today call all comic books and animated cartoons, generically, Miky Maous (ÎœÎ¯ÎºÏ… ÎœÎ¬Î¿Ï…Ï‚).

Or how Americans call all facial tissues "kleenex". Or all disposable bandages "band-aids". Or all photocopiers "xeroxes". Or how any Web search is called "googling".
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
icarus wrote:He says that the conquerors of Constantinople got there by asking "where's the city," and kept getting the reply ending in "to the city" thus that is how he says Istanbul got it's name (I don't think it's true. I think it's K(i)n(stan)tino(bul)is = Istanbul.
It's plausible. The phrase eis tÄ“n polÄ“ " in(to) the city" can get the pronunciation /is tim 'boli/.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

wendy.krieger
wendy.krieger
Vol I Chapter IV of "Dickson's History of Number Theory" carries the title of an equation we call "Euler's Quotient". Even the Wikipedia page on Euler's quotient is more informative than Weiferich's primes.

One of the reasons which I am not really fond of 'honour-names' is that it leads students into suspecting that such things are described by that person, this can be a long and blind search.

There is a thing called the "Wythoff Notation" which i gloss as the "decorated schwarz triangles". Wythoff was involved in providing the basis of the dots on Coxeter-Dynkin symbols, but had nothing at all to do with the notation. In fact, he used Stott's notation, and is more in keeping to call the t_x,y,z the wythoff notation.

One could look at http://z13.invisionfree.com/DozensOnlin ... topic=1379 "Why I don't like maths" particularly the third post.