Wieferich Primes And "sevenites"

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
This thread concerns itself with the use of Wieferich primes w to determine factors in ratios merely by their base b expansions.

(See this earlier thread on sevenites 12 September 2012 for further explanations of Wendy's term.)

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
Wieferich Primes

Those interested in the mathematical definition of a Wieferich prime can educate themselves here, with a "friendlier" explanation here. Of interest to us are base b Wieferich primes 2 <= w <= lim, the limit being some reasonably small prime that might be useful to us. The smallest w for base b is given by OEIS A039951.

The practical application in base b of w and certain small integer powers e of w is the sharing of a recurrent period, i.e., the length of the group of digits after the radix point that is repeated. All Wieferich primes w are coprime to b thus they are purely recurrent as denominators.

Let's look at 1/5 and 1/25 in base 18. The former is ".3a37..." and the latter is ".0ch5...". Now 1/5 is maximally recurrent (which is seen as a disadvantage); this aside, we see that 1/25 is no worse in terms of period. (The base-b expansion of the reciprocal of a prime q coprime to b is maximally recurrent if its period is b - 1.) A better situation subtends for 1/7, 1/49, and 1/343, the smallest three positive integer powers of 7 in base 18: these are ".2a5...", ".06b...", and ".00h" respectively. These powers share the relatively brief period of 1/7 in base 18.

In octal, 3 is a Wieferich prime: 1/3 and 1/9 are ".25..." and ".07..." respectively. This makes working with thirds handier than it otherwise would be, despite the fact that 3 is coprime to 8.

Perhaps the very most stunning case of a Wieferich prime's utility occurs in decimal regarding w = 3. Decimally 1/3 and 1/9 are ".3..." and ".1..." respectively. The prime divisors of 10 are 2 and 5; with the minimally recurrent 3 supplying a minimally recurrent (by definition, as 3^2 = 9 = b - 1 or the decimal "omega"). Despite the fact that 10 has "skipped" the prime 3, decimal users enjoy some transparency of the divisibility of an arbitrary decimal integer by 3 and 3^2. Everyone is familiar with the decimal divisibility rules pertaining to 3 and 9. To determine if a number n is divisible by 3, simply total the digits of n. If the total is divisible by 3 then n is also divisible by 3. This divisibility rule is perhaps the simplest such rule outside of examining the number of zeros at the end of a number to determine if n is divisible by the base. This "infilling" of accommodation for the "skipped" factor 3 makes decimal more useful than it otherwise would be.

The following table gives Wieferich primes w with indices of 30 or below, i.e., primes {2, 3, ..., 113} for bases 2 through 120 inclusive:

Code: Select all

2&#58; &nbsp;
3&#58; &nbsp;11
4&#58; &nbsp;
5&#58; &nbsp;2
6&#58; &nbsp;
7&#58; &nbsp;5
8&#58; &nbsp;3
9&#58; &nbsp;2, 11
10&#58; &nbsp;3
11&#58; &nbsp;71
12&#58; &nbsp;
13&#58; &nbsp;2
14&#58; &nbsp;29
15&#58; &nbsp;
16&#58; &nbsp;
17&#58; &nbsp;2, 3
18&#58; &nbsp;5, 7, 37
19&#58; &nbsp;3, 7, 13, 43
20&#58; &nbsp;
21&#58; &nbsp;2
22&#58; &nbsp;13
23&#58; &nbsp;13
24&#58; &nbsp;5
25&#58; &nbsp;2
26&#58; &nbsp;3, 5, 71
27&#58; &nbsp;11
28&#58; &nbsp;3, 19, 23
29&#58; &nbsp;2
30&#58; &nbsp;7
31&#58; &nbsp;7, 79
32&#58; &nbsp;5
33&#58; &nbsp;2
34&#58; &nbsp;
35&#58; &nbsp;3
36&#58; &nbsp;
37&#58; &nbsp;2, 3
38&#58; &nbsp;17
39&#58; &nbsp;
40&#58; &nbsp;11, 17
41&#58; &nbsp;2, 29
42&#58; &nbsp;23
43&#58; &nbsp;5, 103
44&#58; &nbsp;3
45&#58; &nbsp;2
46&#58; &nbsp;3
47&#58; &nbsp;
48&#58; &nbsp;7
49&#58; &nbsp;2, 5
50&#58; &nbsp;7
51&#58; &nbsp;5, 41
52&#58; &nbsp;
53&#58; &nbsp;2, 3, 47, 59, 97
54&#58; &nbsp;19
55&#58; &nbsp;3
56&#58; &nbsp;
57&#58; &nbsp;2, 5
58&#58; &nbsp;
59&#58; &nbsp;
60&#58; &nbsp;29
61&#58; &nbsp;2
62&#58; &nbsp;3, 19
63&#58; &nbsp;23, 29
64&#58; &nbsp;3
65&#58; &nbsp;2, 17
66&#58; &nbsp;
67&#58; &nbsp;7, 47
68&#58; &nbsp;5, 7, 19, 113
69&#58; &nbsp;2, 19
70&#58; &nbsp;13
71&#58; &nbsp;3, 47
72&#58; &nbsp;
73&#58; &nbsp;2, 3
74&#58; &nbsp;5
75&#58; &nbsp;17, 43
76&#58; &nbsp;5, 37
77&#58; &nbsp;2
78&#58; &nbsp;43
79&#58; &nbsp;7
80&#58; &nbsp;3, 7, 13
81&#58; &nbsp;2, 11
82&#58; &nbsp;3, 5
83&#58; &nbsp;
84&#58; &nbsp;
85&#58; &nbsp;2
86&#58; &nbsp;
87&#58; &nbsp;
88&#58; &nbsp;
89&#58; &nbsp;2, 3, 13
90&#58; &nbsp;
91&#58; &nbsp;3
92&#58; &nbsp;
93&#58; &nbsp;2, 5
94&#58; &nbsp;11
95&#58; &nbsp;
96&#58; &nbsp;109
97&#58; &nbsp;2, 7
98&#58; &nbsp;3
99&#58; &nbsp;5, 7, 13, 19, 83
100&#58; &nbsp;3
101&#58; &nbsp;2, 5
102&#58; &nbsp;
103&#58; &nbsp;
104&#58; &nbsp;
105&#58; &nbsp;2
106&#58; &nbsp;
107&#58; &nbsp;3, 5, 97
108&#58; &nbsp;
109&#58; &nbsp;2, 3
110&#58; &nbsp;17
111&#58; &nbsp;
112&#58; &nbsp;11
113&#58; &nbsp;2
114&#58; &nbsp;
115&#58; &nbsp;31
116&#58; &nbsp;3, 7, 19, 47
117&#58; &nbsp;2, 7, 31, 37
118&#58; &nbsp;3, 5, 11, 23
119&#58; &nbsp;
120&#58; &nbsp;11


Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
The notion of "Wieferich primes" and their related powers is sometimes referred to in this forum as "sevenites". The phrase "prime which divides its own period" is also used. It is perhaps more accurate to say that the prime w shares period &#955; with some of its consecutive smaller powers w^2, w^3, etc.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
It's pretty inane to call these things "sevenites", simply because one person, in their first encounter with these beasts, happened to stumble across one obscure example (seven) pertaining to one obscure base (eighteen, aka sixzeen, aka sesqui•dozen, aka dine).

Imagine if space aliens came down to one obscure town, let's say Helena, in one obscure province, Montana, in one backwater sovereignty, the USA, on one of the smaller continents, and asked one of the native life forms what this place was called. Suppose the befuddled local answered "Helena". Then suppose these aliens insisted on calling all humans "Helenites" forever after, despite being repeatedly informed that "humans" or "Earthlings" would be more apt. Yeah, it's about as inane as that.

About as inane as the Romans (and everyone forever after) calling their trans-Adriatic neighbors "Greeks", simply because the first ones they stumbled into were some obscure coastal tribe that called themselves "Graikoi", when that whole ethnicity in general has always called themselves, and to this day still call themselves, "Hellenes".

I think I'd prefer "Wieferich primes" or "period-dividing primes". There's nothing quite as clarifying as calling something by an appropriate name.
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wendy.krieger
wendy.krieger
Mind you, we have Indians from the sub continent, and Indians from the new world, on the supposition that Colombus reached the East Indies.

Of course, there are Weiferich primes, there are Marianoff primes, Wilson primes, Shanks' primes, and a few others, all pointing to sevenites in various bases and conditions. None of these people garner more than five lines in Dickson's history of mathematics, which spends no less than 12 pages on this subject. Likewise, there is no generic name for these things in Yates' monograph on 'Repunits'. Wells gives five separate sevenites in his 'Curious and Interesting numbers', but associates Weiferich to a passing paragraph in just one (1093 in base 2).

Of course, Weiferich did not discover sevenites: there is some grossary of years before his 1908 paper, and nor did he provide any specific examples not previously known (as Shanks did). His allusion is not specifically to some base, but that the binary sevenites have some essetaric relation with a class of solutions to Fermat's last theorm.

The trouble about using this name (which, as a general form, i can find no printed reference dating before after I wrote about sevenites here), is that it will send students who wish to delve deeper into the subject onto a wild goose chase, after much frustration, a dead end. This is not good science.

The sevenites that provide the name, represent the smallest compound sevenite within reach of hand-calculations: 7 in 18. This is one of the 'algebraic sevenites', derive from equations like 1,3³ = 1,9,27,27; is in base 18, the same as 27,27,27; and as such is the product of two cubes 27 and 1,1,1. You can reach it through base 6, by the equation that 1,1³ = 1,3,3,1 = 9,3,1; hence in base 3,0 we have a cube 1,1,1. Other algebraic sevenites include 1,0,2² = 1,0,4,0,4 => 1,1,1,1,1 (base 3). The only other repunit power known (over 1,1), is base 7 (1,1,1,1).

Of course, sevenites go much further than the self-dividing period thing. Wilson primes of the first and second type, are instances where p^2 &#124; (p-1)!+1 [eg 5], and p^2 Z (p-2)!-1 [eg 11]. There are sevenities of type 'g' (where the smallest primitive root of some p, is g, and the sevenites of type g are those where p² &#124; g^(p-1)-1.

The class-2 isomorphic bases are very base-like structures, but the periods of the primes can divide either p-1 or p+1. These have different gaussian constructions, but things like the density of proper powers, and the density of sevenites is remarkably similar. In this scheme, primes have two different primitive roots, a high and low version. There are sevenites of type h and l, and there are sevenites against all of the iso-bases, although like regular bases, some have no recorded examples (such as j3 = fibonacci numbers). j4, which finds currency in finding the largest of the Messerine primes, the test here is that if 2^p+1 were prime, then it would be an upper long, and this is necessary and sufficient.

Of course, we could just as easily render these primes as Marianoff primes, or Shanks primes, or Wilson primes. Shanks actually found examples. But none of these people provided any deep insight into the theory of sevenites, but list curious features of these numbers.

The main issue here is that Wythoff had little to do with 'Wythoff's notation', as used for the starry uniform polyhedra. Instead, this was a name invented by five mathematicians (including Coxeter), to "honour" Wythoff, and forever to distract students into making errors on Wythoff's actual work. The use of Wythoff in the PG corrects this.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
wendy.krieger @ Jul 21 2017, 08:55 AM wrote:Mind you, we have Indians from the sub continent, and Indians from the new world, on the supposition that Colombus reached the East Indies.
Indeed another inanity of history. (Which is why I've always liked the name Harry Turtledove came up with for the latter ethnicity, in his Atlantis alt-history series: "Terranovans". Alas, people are unlikely to change terminology based on fiction.) But pointing out inanities of the past should be an object lesson in what not to do, not an excuse to perpetuate further inanities.
Of course, we could just as easily render these primes as Marianoff primes, or Shanks primes, or Wilson primes.  Shanks actually found examples.  But none of these people provided any deep insight into the theory of sevenites, but list curious features of these numbers.
Well, "honor" names are equally inane, if the "honor" is actually misplaced. But just because one can rant about those doesn't give "obscure-instance-inaptly-genericized" names a pass. Indeed, such could be seen as a kind of "honor" name, too -- honoring beyond all proportion the peculiar specific example which lead to some "deep insight", rather than the actual insight itself.

Which is why, ultimately, something like "period-dividing primes" is probably the best bet. That's what they _are_, not who found them first, or where they were first found.

As to your other points, perhaps the best reply one can give immediately is that they constitute a Gish Gallop, on top of the usual indulgence in unexplained-idiosyncratic-notations.
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wendy.krieger
wendy.krieger
I use the term sevenite for the general 1/(p²-p) distribution, not just the class-1 base sevenites. I was hoping to give examples, such as the various Wilson-prime examples, and a class where the smallest primitive root of p has p as a sevenite as well. This means that the smallest primitive root of p is not the smallest primitive root of p^2. Generally, this is a case of the second remainder p aligns with a special case for the first remainder.

But like Coxeter's extensions of geometry, I extend bases into more radical examples, such that eg

class 1: base b/a = b^n - a^n : for factorisation ends
class 2: isobase d/c = b^2+a^2 = d; ab = c: then as base b/a.

The integer forms have the denominator as 1. Note that for calculation purposes, the class 1 bases evolve into a class 2 base, but in class-2 the numbers a, b do not have to be integer. They are generally quadratics.

The base-tail notation BbTt, is the last 't' digits of a number written in base 'b'. It's the tail-end of an integer. The "sevenite tail" is an expansion in base 'p', which the base is compared against. The more digits the two tails agree, the more the sevenite is, eg.

01550155 B13T8 sevenite-tail for '5'. 13 has a 4-place period in base 13x+5.
00000055 B13T8 tail for 70. Only 13^2 divides the 4-place period.
00000155 B13T8 tail for 239. The last four digits agree, so 13^4 divides the 4-period.

"Bond" sevenites have a sevenite-tail ending in BpT3 as 0 0 b, reflecting Ian Flemming's character's code number '007'. There are only three bond-sevenites of any type known. The sevenite-tail for B113T3 includes 0,0,68. 113^3 divides 68^112-1.

"Compound sevenites" are any sevenites of order three or greater. The tail agrees with the sevenite place to three or more digits. 13 is a compound sevenite in base 239.

Wilson sevenites of the first and second kind, do not 'divide periods', but have the 1/(p^2-p) distribution. These correspond to dividing p^2 &#124; (p-1)!+1 or (p-2)!-1.

Primite-root sevenites are where the smallest primitive root of p, ie 'g' has the prime as a sevenite. This is not a dividing the period. Regular bases have one primitive root "g", class-2 bases have 2 primitive roots 'h', 'l'. Sevenites are known in all three cases.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
This doubles down on your Gish Gallup, tossing out many terms and notations that are nowhere near being explained sufficiently so a reasonably-intelligent person with a fairly good background in mathematics can get more than a vague notion of what you are actually doing:
... the general 1/(p²-p) distribution ...

... primitive root of p ...

... first remainder ... second remainder ...

... class 1: base b/a = b^n - a^n    :  for factorisation ends
class 2: isobase d/c = b^2+a^2 = d; ab = c:  then as base b/a....

... It's the tail-end of an integer. ....
How is 0155055 the "tail" of 5? How is 00000055 the "tail" of 70d? How is 00000155 the "tail" of 239d? Absolutely no explanation where these "tails" come from. It is not even clear what is the "base" being used. The B13T8 seems to imply it's base 13d, but since you mention "base 13x+5" and 5, 70d, 239d are all instances of that formula, are those supposed to be the bases? The whole discussion is incoherently garbled.

Each of these ideas/tools/formulas could be the subject of a whole post or thread. Many have been mentioned in the past, yet never really adequately articulated. But by this point, whatever glimmer of interest readers might have had in these ideas is rapidly evaporating ...
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)

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
I've used the term "sevenite" merely because it has been used in the forum not only by its progenitor but by me, thinking that it were a standard term. Also, it is likely or possible that the term "sevenite" is not fully "coterminous" with the mathematical standard term "Wieferich prime." Wendy has demonstrated that her term has more dimensions than the dimension of a "prime that divides its own period."

Personally I think that "prime that divides its own period" is not an apt "vanilla" moniker for "Wieferich prime", or the concept of such that would be of interest to people thinking about the utility of number bases. The prime 7, for instance, does not divide its own period (3) in base 18. Instead, I would say that the base-18 expansion of the unit fraction with 7 as denominator shares the same period with the next two smaller integer powers of 7 (i.e., 7^-2, 7^-3). That is a mouthful. Maybe "period-hold primes" but the whole concept of "period" in this day and age just makes adolescents snicker. So maybe "reptend length hold primes". When it comes down to it, using the term "Wieferich prime" out of the box, so to speak, seems to make the best sense.

I won't excoriate anyone for coining neologisms ; ) but I would like to know if one is doing such, so that I don't take it and run with it as if it were the term of art. It is plausible that people, when encountering a concept for the first time (either personally or actually discovering the concept) may assign an errant name that somehow sticks. Of course we want to prevent that if we could, either by researching what the term of art really is, or by ensuring that our name is the very best name for that concept. Whenever we use the term "best" it opens the door for subjectivity. So we'll have to trust that whoever names the concept is doing their best ( ; ) best being subjective).

I don't mind the name "sevenite", but we are merely interested in the aspect as it pertains to shortening the periods of base-b expansions of small powers of primes q coprime to b, where the period of 1/q itself in base b is already pretty short, especially if p_1 < q < p_k, where p_1 is the least prime factor of b and p_k is the greatest prime factor of b, and b >= 2 is itself a highly divisible and relatively small integer. Pretty tight constraints. Wieferich prime 83 base 99 wouldn't seem too relevant for most people's purposes, but Wieferich prime 3 base ten is a critical feature of decimal, as 2 < 3 < 5, and 1/3 has period 1 and shares that with 1/9. That is just about the smallest such case we can identify, wherein the Wieferich prime q completely fills the "gap", has minimal period, and pertains to a rather small base b. This would seem to imply that the importance of Wieferich primes is rather minor and the case of 3 in 10 is the very most significant case we might discuss, with some relevance given to the situations at 18, et al. It certainly is not on par with other considerations like the magnitude of b itself or the divisors of b, but it does merit mentioning even only for its wild effect on decimal. Most of the other aspects of "sevenites" are mathematical curiosities interesting in and of themselves, but won't help one bake a cake or lay up a wall, i.e., their practicality or relevance to everyday existence is paltry or nil.

Dozens Demigod
Double sharp
Dozens Demigod
Joined: Sep 19 2015, 11:02 AM
Honestly, the only cases where this actually makes any difference to the average person are where the Wieferich prime is:

(1) relatively small, so that there is still a reason to worry about their powers, which actually seems to restrict things to just 2 or 3;
(2) also the alpha or the omega, so that it actually is beneficial (it does not do much good if 5 is maximally recurrent, and having 5^2 have the same short period does not sweeten the pill very much).

When we consider that having 2 as a Wieferich prime means that your base is odd, which is basically a non-starter, the only cases that really impact the immediate feel of a base seem to be 3 in octal and decimal, and of course the effect is stronger on decimal than octal. It is this that vaults 8 and 10 up and makes at least the latter perhaps the only serious challenger to 12 as the best base for humans.

Some cases look mathematically interesting, like the alignment at 99. But using 99 this way in an application seems to need some way to use the arithmetic of base 99, and while 9-on-11 alternation does work, it almost seems like it is easier to just use divisibility tests in an easier base - because if you want to test for the square of a prime, it is rather cheaper to test for the prime first if it is easier.

Obsessive poster
Obsessive poster
Joined: Sep 10 2011, 11:27 PM
[quote="icarus @ Jul 21 2017, 02:59 PM"]The prime 7, for instance, does not divide its own period (3) in base 18.[/quote] Okay, so a more careful definition of a Wieferich prime is not that it divides its [i]period[/i], but rather that it divides the [i]periodic part[/i] of the digital expansion of its reciprocal (a la OEIS sequence [url=https://oeis.org/A036275]A036275[/url], but generalized to any base). So would "periodic-part-dividing prime" be more apt? Isn't the test for the Wieferichness of 7 in base sixzeen as follows? (Using underscore rather than overbar for repetends.) So seven actually goes one step beyond just being a Wieferich prime, in that it divides both its own periodic part and the periodic part of its square.
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)

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
I am saying this about nomenclature not to be a stickler, but merely to point out that it is easier simply to "buy" the going term of art and employ it, especially for a quality Double sharp aptly shows is mostly unimpressive and an esoteric circumstance.

It is possible I might be missing something regarding Wendy's "sevenites" or Wieferich primes but I do think it is keen to presume as Double sharp has outlined, that in most cases these entities are merely "icymi" sort of tidbits and not the stunning solutions to all worldly problems that, say, multiplication is.

To be sure, Wendy's "sevenites" probably do have an idiosyncratic and to the average Joe, esoteric, application. For most practical considerations the properties of the other flavors of "sevenites" are likely not "practical" considerations but interesting details. (Correct me if I am wrong or missing something stunning).

The thread is simply trying to get to the bottom of what we might be missing, and to give voice to what the things are. The conversation is good.

Dozens Demigod
icarus
Dozens Demigod
Joined: Apr 11 2006, 12:29 PM
(This discussion immediately followed the OP of this thread and was followed by this segment before the 28 July 2017 split. It was believed by the admins that this better organizes the forum and that these topics merited their own discussion space. Apologies to all for confusion.)

wendy.krieger
wendy.krieger
You can see the underlying theory of sevenites from the ordinary repeaters.

In base 2, 1/3 is 0.010101... This is a 2-place period. 1/9 has a six-place period, in the form 0.000111000111...

In Octal, you see that these concentrate: 1/3 is 0. 010 101 = 0.25, and 1/9 is 0.000 111 = 0.07.

However, you can always 'put' the base adjacent to some power or product, to get sevenites, and the alpha-omega sevenites never roused any real interest. Instead, it required 'deep' sevenites, with periods of three or greater to kick this off.

The sevenites in decimal and dozenal, are too far burried to arouse any real interest, but those in eighteen are actually quite accessable, and also deep. The compound sevenite in base 18, is quite accessable, and perhaps the only known example of A3 as a cube. (A3 is the new factor in RRR, after A1 = R is taken out).

The examples of a concentrated sevenite is seen in 32 and 128, where

1/5 = 0.00110 01100 11001 10011 = 0.6 12 25 19 b32
1/25 = 0.0001 01000 11110 10111 = 0. 1 8 30 23

1/7 = 0.0010010 0100100 1001001 = 0.18 36 73 b128
1/49 = 0.0000010 1001110 0101111 = 0.2 78 47

You can see that these sevenites arise purely from the grouping of binary digits into groups of 5 or 7, and thus both 5 and 25 appear with a 4-place period and 7 and 49 yield a 3-place period in base 128.

The prime powers yield to the same rules as the primes themselves in Fermat's little theorm. That is, every base which 7 does not divide, 49 has a period that divides the totient of 49, which is 42. There is, as with 43 (which also has a totient of 42), a 'primitive root' (three in both cases), whose first 42 powers have different remainders against these numbers. This number is the 'index'.

Code: Select all

             43                               49
1   7   7    7   7   7   7        1   7   7   7   7   7   7
=============================     ==========================
1  21  11   16  35   4  41  &#40;1    1   8  15  22  29  36  43
37   3  20   33   5  19  12  &#40;6   31   3  24  45  17  38  10
36  25   9   17  13  15  14  &#40;3   30  44   9  23  37   2  16
42  22  32   27   8  39   2  &#40;2   48  41  34  27  20  13   6
6  40  23   10  38  24  31  &#40;3   18  46  25   4  32  11  39
7  18  34   26  30  28  29  &#40;6   19   5  40  26  12  47  33


This is the period table for 43. The powers of 3 run in steps of one right, one down, toro-cyclic. The periods of the numbers are then the product of the row and column headers, so 12 (last column, second row), has a period of 42.

The right hand side shows the period-length for 1/49. Each number on a row has the same remainder modulo 7, but only the first column has a period-length dividing 6, and thus has a sevenite of it.

Calculating Sevenites

Bases in the main do not have a lot of sevenites. The table of listed sevenites per base rarely exceeds eighty characters, and many times much less. Some bases, like 18, have a lot, but even 5, 7, 37, 331, is spreading the friendship a bit thin.

The first way is to calculate some 'g', or primitive root of the base, and then raise it to g^p mod p^2 = G. Then you calculate G mod p.

2187 mod 49 = 31. The powers of 31 are then as listed above, giving a sorted sevenite table of

7: 1, 18, 19, 30, 31, 49.

The second method is to use the second-remainder: g^(p-1) div p mod p. This is the Euler Quotient. It suffices to simply calculate the primes one is interested in, since these function as logrithms. (they correpsond to the tens digit in 11^n). These numbers are written in base 7, the sixth power of the first number, the reduced value is the second remainder or 'tens' digit.

For 7, we see 2 = 121 -> 2, 3 = 2061 => 6, 5 = 63361 => 6, 11 = 21025621 => 2.

To demonstrate that 18 has a sevenite 7, we see 18 = 2*3*3 -> 2+6+6 = 14, thus 7 divides 14, and hence 7 divides its period base 18.

The third method is to calculate the 'sevenite tail'. These are the last digits in base p, which corresponds to the sevenite ending in that digit. In the table above, it corresponds to the first column, viz

01, 43, 42, 66, 24, 25.

or sorted by the last digit: 01, 42, 43, 24, 25, 66.

This method is only used of small primes. The largest compound sevenite seen in nature (ie from an extract of a range, is the regular 113, and the isosevenite 6617.)

A rexx script was then used to crawl through this data (the zip files sits on a DVD), looking for 'bond sevenites'. The classic bond sevenite has the sevenite-tail of

0 , 0, 68, base 113.

The name derives from Ian Flemming's character "James Bond", agent 0 0 7.

The massive table i produced in 2006, called 'forty smallest sevenites', used the first method, with some adjustments to accelerate the process.

1. The method used was insertion sort. To make this work, we did not consider any value unless it was smaller than the largest number of the array. This array was spiked with 60p in all 40 values.

2. We considered both if it was less than 60p or greater than p(p-60). This cuts the calculation time down, since if s has p as a sevenite, so does p²-s.

These calculations were done to 2.5 million, according to the decimal reckoning. The actual batches were done in primes inside the same thousand, twelftywise.

wendy.krieger
wendy.krieger
An interesting thing is that both 53 and 67 has 47 as a sevenite, and therefore, all numbers of the form $$53^x 67^y$$ have 47 as a sevenite.

The downside, is that there can only be at most 46 examples of these numbers between any two multiples of $$47^2$$.

 Posts 33
Casual Member
Einmaleins
Casual Member
Joined: Feb 12 2018, 06:58 AM
icarus wrote:Most of the other aspects of "sevenites" are mathematical curiosities interesting in and of themselves, but won't help one bake a cake or lay up a wall, i.e., their practicality or relevance to everyday existence is paltry or nil.
Like other number curiosities they are interesting to look at if one is searching for patterns in the bases. No practical use, maybe, but many litle items like these in number theory amuse us. I am still looking for Fermat proof. Something to think about when on a long delivery, especially with all that snow.

Dozens Disciple
Shaun
Dozens Disciple
Joined: Aug 2 2005, 04:09 PM
Einmaleins wrote:
Like other number curiosities they are interesting to look at if one is searching for patterns in the bases. No practical use, maybe, but many litle items like these in number theory amuse us. I am still looking for Fermat proof. Something to think about when on a long delivery, especially with all that snow.
Many discoveries have been made by simply searching for patterns.
I think you might need a to travel over a very long delivery route before you reach that proof ...
I use the following conventions for dozenal numbers in my posts.

* prefixes a dozenal number, e.g. *50 = 60.
The apostrophe (') is used as a dozenal point, e.g. 0'6 = 0.5.
T and E stand for ten and eleven respectively.