Baroque conjectures

Let me address the views (guesses) on the four questions, about the perfect and baroque numbers, listed in the previous post:

• everybody believes that there are infinitely many (even) perfect numbers, because everybody believes that there are infinitely many Mersenne prime numbers p (i.e. prime numbers p such that 2^p-1 is a prime too);

• it seems to me that some (many? most?) specialists believe that there are only finitely many baroque numbers which are not perfect; some even think that all baroque non-perfect numbers are already known (no way!). Myself, I believe that there are infinitely many of them but finitely many for each baroqueness coefficient;

• nobody believes that there is any odd perfect number;

• I think that nobody believes that there is any odd baroque number; I don't either but I am not so sure :-).

I am interested in the race toward finding more baroque numbers.

Back to baroque numbers?

The classical, standard name for my term baroque numbers is pluperfect or polyperfect
numbers or similar. A natural number n (n = 1 2 ...) is called baroque when the sum sd(n) of all natural divisors of n is divisible by n; let's call

brq(n) := sd(n)/n

the baroqueness of n. When a baroqueness of a number is two, brq(n) = 2, then it is called perfect. The smallest perfect number, known since the ancient times, is n=6. Indeed, in this case

sd(n) = 1+2+3+4+6 = 12 = 2*n

Questions about the baroque numbers are among the oldest, open (unresolved) problems of the whole mathematics. They are difficult and, today, somewhat isolated from the main developments, hence they are not as popular among the best mathematicians as they used to be in the past, when Euclid, Fermat, Decart, Euler and others were interested in them. Let me list the main open questions:

• are there infinitely many perfect numbers?
• are there infinitely many baroque numbers?
• does there exist an odd perfect number?
• does there exist an odd baroque number > 1?

Of course n=1 is the only number for which brq(n) = 1.

The Euclid-Euler tandem (:-) proved that an even natural number n is perfect if and only if there exists a (unique) natural number p such that the following two conditions hold:

• 2^p - 1 is prime;
• n = 2^p-1*(2^p-1)

It is well known and easy to see that when 2^p-1 is prime then so must be the exponent p itself.

Another old result (an ancient observation) is that brq(120) = 3, i.e. 120 is a baroque number, and its baroqueness is 3.

Everyday language

The common goal of a conversation is to get to the point, without an unnecessary worrying about the precision in relation to the side issues, which are mentioned only casually. Possibly, the side items were mentioned without even meaning them as something to be dwelt upon anymore. We are humans and not robots, hence we add occasionally this or that.

The lack of precision of the everyday language is its strength, but strengths and weaknesses tend to walk together. The everyday language will be abused easily when there is no good will on one or more of the conversing parties. And one of the ways is to zero on the side issues, when the other party have not even dreamed about actually discussing them and having their each word scrutinized in a hostile way.

For example, in the everyday language we often don't make it clear whether we talk about the total set or about the existence of just one member of a set, as in the phrase "young people tend to be careless" - are ALL young people assumed to have a tendency to be careless or just SOME of them? The speaker most likely didn't care about it at the time of saying it. But the precision police would give that speaker hard time. The surprised and shocked and frustrated by the attack speaker would possibly defend its pronouncement, would get trapped into new situations, and would be doomed. If you attack someone who didn't mean any battle, then the person will say things ad hoc, and you will prevail, and the communication will lose. You will always find something, you will always generate some "faults" in the other person, and you will feel so righteous.

Add to this some relaxed phrases, some light jokes, an occasional not politically correct wording, and the person allowing itself of such a luxury will be doomed, there will be so much material against the unsuspecting soul. The winner prevails, but how intellectually and emotionally poorer is the winner's life! And how meager are such "victories"!

Interpersonal (mis)communication, public and private

There is so much of miscommunication. A lot of it because people simply don't like one another, in which case the truth does not matter. There are also power games, ego games, psychological insecurities, etc., both in public and in personal exchanges. The difference of experience, of knowledge, of understanding, of class, between the participants also play a big role. Even when two people care about each other, the insecurities, the games, and similar, may still spoil the communication.

In adversary exchanges there is no point to even worry about such things. One should and often does address the public and not the adversary. Most of the time there is no chance to affect the adversary. For these reasons I have developed a flexible, neutral format for Internet discussion groups (it's an application of my Art of Agreement), but that is a topic for another thread.

Below, I'll provide an example of a (fictitious but realistic) conversation which went down the drain.

---

***

A. and D. talk about the treatment by different computer languages of the upper/lower case in the names of variables and similar. First languages had upper case only (e.g. Fortran). A. and D. remember that other old languages allowed for the either case; the lower and the upper case of any letter was treated as equivalent. Thus Mary mary MARY maRy mARy etc. would all stand for the same variable. This feature was widely used by programmers to enhance the readability of their programs. In particular A. was fond of the especially consistent and useful way this was done at the company for which A. had worked years ago.

Then D. mentions that a new option is offered by the wiki engine. According to this convenient in the wiki contest convention, wiki is sensitive to the case of all letters in the path except for the very first letter: you may write Albert_Einstein or albert_Einstein, and it will amount to the same link, but Albert_einstein and ALbert_Einstein will be different (and most likely each will be just a mistake).

D. knows A., hence he said carefully wiki engine. There are several wiki and wiki-like engines, while there is only one default wiki engine, the one that powers the English wikipedia (and several other wikipedias and similar portals, while there are also wiki portals which use different engines). All these things both A. and D. know well over the years of being somewhat interested in the topic (and D. being even active in some wiki portals; and each of them knows that the other one knows). D. was careful to say wiki engine, while it would be an exaggeration to spent more words on this precision. Nevertheless A. objects that not all wikis are alike, and asks: how do you know that all wikis are like this? D. answered that he knows only about three of them, including the English wikipedia. The conversation is ridiculously going along this line, with A. accusing D. of being a liar! According to A., the other person claimed in the beginning that ALL wikis are supporting the mixed convention. D. knows very well that it was not the case, of course not. So D., irritated, responds that it is A. who is the "liar". In a sense, D. even believes that this is the case – that A. is manipulating the conversation instead of allowing the pleasure of joint exchanges of knowledge and reminiscences.

Super! :-) Now, what is the mechanism of such disasters? It seems that A. has the need of being right, and being the only one who is right. Thus others have to be wrong. The conversations for A. are trials where he is the prosecutor, the investigation officer, the judge and the jury. While D. naively treats the conversation like it were about the upper/lower case option, it turns out that the conversation is about D., that D. has to defend in true or imaginary detail each of D.'s casual pronouncements -- surprise, surprise! A. believes in being always right, that others are sloppy, illogical and stupid. A. is sane enough and has the necessary self knowledge about A. not being an Einstein, that A. has not won a Noble prize nor the wealth of Bill Gates. Thus to have its self-respect A. needs to put others down. A. will go so far (and with a strong conviction! -- ridiculous as it sounds) as to call others liars, where it is A.'s inability to listen and to stick to the topic. One may, like A., have difficulties to concentrate and catch all the fine points but then one should allow for the margin of not getting them, and of not understanding everything all the time. A good conversation requires patience and trust. And first of all, a conversation should not be a trial on which one puts the other person, who never got any court invoice. When a conversation is strictly about a computer language option of upper/lower case , with neither partner having any material interest in the topic (just intellectual curiosity), A. has no right to switch to an investigation of the D.'s personality qualities (say, to true or imagined D's inability of expressing things precisely) -- that's absolutely wrong.

The basis for such disasters is psychological, but there is also a tool (a method), which A. uses with a practiced agility to achieve such spectacular negative effects. The A.'s method is grounded in the common usage of the everyday language (as opposed to scientific language). See the next post.

Dabanese 2008. Syntax.

A dabanese reader/writer, dabaner for short, has to know only four notions and the simple syntax rules. The notions are:

• ideogram
• unordered daba phrase
• ordered daba phrase
• accent (subject)

Now the rules:

1. ideogram is a daba phrase;
2. a finite sequence of daba phrases enclosed between braces, {...}, is a daba phrase – it's called an unordered phrase;
3. a finite sequence of daba phrases enclosed between parentheses, (...), is a daba phrase – it's called an ordered phrase;
4. a daba phrase enclosed in brackets, [...], is a daba phrase – it's called an accented phrase or a subject;
5. all daba phrases are obtained by a finite application of rules 1-4.

All elements of dabanese should be separated by white spaces (or they may be separated by some other graphic device).

Phrases { A B C } and { B C A } are in principle equivalent, they have the same meaning in dabanese, the difference is at the most artistic and similar. When you quote a daba text then you may change the order of subphrases in the unordered phrases (unless there is a claim of the completely exact quoting). In particular it is legal to change the order of subphrases in unordered phrases when quoting in legal situations. On the other hand you must preserve the order of subpphrases of the ordered daba phrase, when you quote them even in informal situations or otherwise it is not a quote. Any change of order in ordered phrase is likely to result in a drastic change of its meaning. Claiming to be quoting when changing on purpose the order in ordered phrases would be cheating, while an inadvertent change would be an irresponsible sloppiness.

Rules 2. and 3. applied to the empty sequence give us phrases { } and ( ), which stand for "nothing" or for emptiness, etc. There will be dabanese dictionaries but there always will be the room for the customary poetic interpretation of the text by the dabaners, depending on the context, just as in the natural languages. For instance, [ hmn ] phrase tells us that human is the subject (of the respective portion of a daba text), as in { ( ) [ hmn] }, which points perhaps to a human who does not represent anything, who is totally uninteresting. Now let's consider phrase [ [ hmn ] ], with a double accent. To me it means that the topic is the attention on human, while I am afraid that to many people it will mean an extra strong accent on human. Thus daba phrase { ( ) [ [ hmn ] ] } means to me something like "an empty (useless) attention on human", while to others it may mean that especially all or some or particular human is silly (more silly than non-humans or other humans), with the extra emphasis on the group which is meant. Possibly, the same phrases will mean different things to different people.

Dabanese cannot and does not attempt to define every reading of the dabanese text. For instance someone may talk about arithmetic calculations using the infix notation ( 2 + 5 ), and another may use the postfix ( 2 5 + ). it is up to dabaners to understand each other. In the given text they should use the ordered phrase in each case. But if they want to make sure to be understood they may add an extra description: let ideogram rPn stand for the reversed Polish notation. Then one may write { rPn ( 2 5 + ) }. Perhaps the best is to write it as follows: { 2 5 [ + ] }. Now it is pretty clear. The subject is sum, but it is a sum described by 2 and 5, so it is 7. Then phrases { 2 [ + ] 5 } and { [ + ] 5 2 } mean the same, i.e. 7. A different meaning would have a phrase like ( [2] + 5) – it would mean something like: 2, to which 5 was/is/will be added.

A daba phrase without any subject (i.e. when none of its subjects are accented; even when a subject of a subject may be accented) are called lists – there are ordered and unordered lists. A daba phrase may have more than one accent but it is not advised. A strict daba phrase should have at the most one accent. If you want to have more than one then do it for instance as follows: { A B [ { C D } ] } rather than by { A B [ C ] [ D] }, but it's up to you.

Dabanese 2008. Pigeon ideograms.

(I see the "New Post" user option, but not "Edit" user option - how frustrating since I want to correct typos spontaneously, when I see them).

I'll write about dabanese from scratch. Writing from scratch is my weakness. After a break I am incapable to just continue.

I'll start to create a dictionary of ideograms, still in their pigeon form. For the sake of convenience I'll consider all English and Polish nouns, in their main form, and noun like words, as pigeon ideograms. However, some ideograms will start to have a legal status. At least conceptually (not graphically). Here we go (ideograms in the same line are synonymous):

• := – ideogram of definition.
• { {} := { } } – this dabanese definition tells us that ideogram {} stands for nothing or Polish nic. Actually, () and nic are synonymous ideograms for nothing, i.e. each of them is legally equivalent to ideogram {}:
  () – synonym of {};
  nic – synonym of {};
  
• daba – the ideogram for everything, as well as for the (universal) data base daba:
  ∀ – synonym for the ideogram daba.
  
• self – ideogram of the reference of the source of the given dabanese phrase.
  się – synonym of the ideogram self.
  

I need to go now. Thus let me just collect cleanly the dictionary so far, without explanations:

• :=
  
• {} () nic
• daba ∀
• self się

Let's remember that ideograms, as well as all records of daba, are partially ordered in daba (regardless of being primary ideograms or derived ideograms; the position of data record in daba sheds additional light onto its meaning). Now I need to go.

2008-June

I am slowly, a bit at the time, learning about the details of this blog's logistic (of the user interface). E.g. I just saw the "new post" menu option at the top of this blog window. Such small details can keep me back. I am hopeless.

I ran into (Artur P)'s blog. Somehow it had a very positive influence on me, made me come back here too, and to take another look at my present situation. Perhaps, with a bit of discipline, I can still squeeze from myself a bit of consistent, constructive effort. My main problem is, or I should mbeliwve that it is psychological. Since Biblical time people understand the necessity of Sabbath or Sunday, and of the holidays, while for years I have none of them. However, iof I somehow fool myself into pretending that I have some, then perhaps I will get going a bit instead of having excuses. Chess players are especially good in making excuses. In this respect I am by far a world champion (otherwise I am another poor, club level chess player).

As a first step let me write some of my present projects and activities (besides assisting my father):

• I participate in a private Internet forum for my school classmates. There are fewer and fewer of us left. The forum has strictly sentimental value. But so what? Isn't it about being humans (for better or worse), and not just about achievements etc? There were really only 7 of us active (on and off). Now it's 9. Others are lurking or not involved (while formally all classmates are automatically among participants, if their email is known).
• I have started to write, in TeX :-), a consecutive article for Delta. This time about the magic squares. I have go back to it.
• I should truly concentrate on my dabanese language, especially, that I got Artur interested (at least for the moment).
• I should go back to writing about my "The art of agreement". In particular, I have written recently 4 installments in Polish for an electronic zine, and I should continue. Somehow that zine is not making the writing process attractive. I should still continue.
• A reincarnation of a poetic Polish portal is in making. Somehow participants are passive. The main force of the project, PŁ, set up a deadline around June 20 something. I should finish writing the statue of that portwal without worrying about inactivity of others. If that portal did happen I would feel real good, hopefully I would write there my more complete view of poetry (it goes back to the old views on poetry). I wish such a nice portal existed in the past, not just for poetry but also in other domains - the quality of my life (and of others, I believe) would be higher.

Well, enough, I better stop now. Sure, I have other projects too, infinitely many of them :-), be it public projects like education or private like physical exercise, better eating and sleeping habits, etc. Unrealistic. On the top of it another infinity of mathematical projects. Sounds crazy but I am too lazy too be crazy. I simply wish I had my peace of mind to learn some profound pieces of mathematics. unrealistic, and this time I feel sad. I would need good conditions to truly get into hard mathematical problems. I don't think that disciplpine alone is enough. I would need discipline anyway, even under favorable circumstances.

---

Artur's blog is interesting. In particular, he has poetic photos of the Cracow's suburbs, Kazimierz & Pogórze.