Not an auspicious start. The on-line editor or rather the software system has decided that I was using the editor without relogin for too long. So, upon the saving operation the system asked me to relog, promptly destroying my file, bye-bye.
There are many advanced blogs which are read by a lot of readers. I admire them, both the authors and the readers. But my own blog is just a diary mostly devoted to this Everything 2014++ site.
I started with the opening page. It includes a subject index which at this moment is ficticious. But in a moment I will replace the announcement of this blog by its link, and that's the beginning. (Indeed, the link is already there--2014/03/23, h.00:15am).
I am trying to have a rich net of links within Everything 2014++. In particular, I already have anchored links to Note 0 and this Note 1 (to have these two links is silly but good for the testing purposes, hey!).
Yesterday (I started this site near midnight), after posting the top level of the subject index, I doubled it as Index (details). Then I expanded the Art of Agreement and the Mathematics list items. The AofA already has a non-fictitious link Concrete suggestions. It leads to an actual file which at this time indeed contains 3 suggestions (the last one is not written in a complete way but even now its meaning should be clear enough). I plan to add also certain concrete small scale inventions. I also need (plan) to have an index for Concrete suggestions.
I learned about including Polish html diacritics, see HTML codes, .... All I need to do is to include line
<meta http-equiv="content-type" content="text/html;charset=utf-8" />
in the HTML header, nice!
It's 5:52am. I didn't start here mathematics yet. At least I added index to Concrete suggestions, and moreover I have added a separate file Simple invention but as a part of the topic Concrete suggestions. Thus I've disclosed two simple inventions:
I decided yesterday, around the noon, to forget everything else but to work on 2014++. I managed to forget about everything else but also about 2014++, how sad! (I do have other things to do too, and I procrastinate about everything).
I have two approaches to working on the products of the initial intervals of primes. I could warm up with predictable results or I could try a harder result right away. In the past I would risk an ambitious route. But I can hardly do much these days so that I'll play a bit, have fun. I can prove initially a few linear estimates which start from a constant which depends on the linear coefficient. Now, to get something real, i.e. better than linear, I should pay an attention to those constants, and the sequence of those so-to-speak constants (:-) would provide a better estimate. But first I should be fluent about those semi-routine/semi-fun estimates.
Now I have mathjax $\LaTeX$ script, and Polish diacritics Ą Ć ... Ź Ż ą ć ... ź ż (i.e. I made a respective addition to the html header).
Tomorrow an errand, and then I go forward full speed. Right, sure. I am destined to procrastinate forever :-).
I should connect an old file, a miniature, to Number Theory. It's against my usual way, I tend to start everything from scratch. I was also trying to call ipage. I wanted to ask them about a daba base package available here, and also some formal questions.
So far the count of my blog notes and the respective days goes in parallel. I don't want to skip a note on any day, which would break the correspondence between the notes and the dates. I'd rather occasionally two or more (truly separate and non-trivial) notes during the same day.
My proof of any of my existing or planned so far prime product theorem features a somewhat strange element of a proof by a contradiction. (One time I somewhat reduced the a contrary argument and the result got... weaker). Nevertheless, according to my old view (philosophy :-) I should formulate the main result (still to be created) in a positive manner, so that the a contrary argument would be reduced to a minimum, just to obtain a customary formulation.
Let's do a bit of a discussion right here. Let
$$P:=\{2\ 3\ 5\ 7\ 11\ \ldots\}$$
be the set of all (natural) primes. Next, let
$$\Pi_x := \prod\{p\in P: p\le x\}$$
for every positive real number $x$.
Cont. A well known by hard to find in literature theorem states that:
$$ \Pi_x \sim e^x $$
This result is roughly equivalent to the prime number distribution theorem. Chebyshev's method or more explicitely Erdos theorem gives
$$ \Pi_x \lt 4^x $$
by elementary considerations. I'd like to attack this problem (which already is a known result) by other elementary approach. Most likely I will get very little, much less than the mentioned classical results but I should have some fun nonetheless. I'll look at this in an equivalent way:
$$ \Pi_{\log(n)} \sim n$$
where the natural logarithm is meant.
Ghrrrr..., the system got better of me again, I lost the last portion of my text. I said that I am losing my heart for the above mentioned mathematical project, that I should decide first of all on my daba+dabanese project. Thus without a delay I should call ipage support about the data base programs available under the ipage roof. I called them a coupple of days ago, and gave up on the account of the heavy volume.
I called, and there was heavy volume of calls again. I gave up on waiting because I was not ready for a fruitful conversation. Or so I thought. While waiting I was checking on MySQL documentation. And how it looks at ipage. I didn't find much but it looks that I need... a new computer! Hm, perhaps the support people could tell me if this is true. Perhaps under Concrete suggestions (where I already have a subfile Simple Inventions) I should also open a file Wishes.
Thus I wish for unseen tables, where a user can initially see only the names of the fields (columns). Since they ae unseen there may be even sixty or more of those fields. Then the user chooses a reasonable number of them for the sake of a display. Then a table with the selected fields would show up. Of course one can selected the primary, and perhaps the secondary, and even a ternary... for the sake of ordering the records (lines) of the table. Each of the two directions of ordering should be available. This ordering feature was actually available already several years ago at wikipedia and other wikis. Blog is ok but I still the wishes file.
Oh, let me go back to number theory :-) The idea is to select a function $f:\mathbb N\rightarrow\mathbb N$ and a constant $N_f$ such that:
$$\forall_{n\in\mathbb N,\ \, n\ge N_f}\ \ \Pi_{f(n)}\ge n$$
The smaller $f$ the better it is. This function should satisfy certain natural conditions like
$$\forall_{m\ n\in\mathbb N}\ \ (n-m)*(f(n)-f(m)\ge 0$$
The whole idea is that first we guess the correct function (the smaller the better), and then we have to prove that the guess was correct.
REMARK If we get a sequence of smaller and smaller functions $f$ then the respective smallest constants $N_f$ will form a function which is smaller than the functions from the sequence. It's not easy to achieve though.
An approach to the problem may work as follows: first there is a constant $M$ such that $\forall_{n>M}\ \Pi_{f(n)} > n$ (sharp inequality!). And then that's what one wants to show. Thus my by a contradiction method.
This time I have decided on two different notes during the same day, today. I intended to write a bit more about number theory (nothing new yet, see above), but also some new daba/dabanese construction/notion. Let me start with number theory. In my notes, say here: Prime products, I've introduced a logical notion:
an implicit inductive argument by contradiction.
It works as follows: the inductive statement $T_n$ is covered by two cases $A_n\ B_n$. Thus we prove $T_{n+1}$ under each of the assumptions. End of proof. However, the case $B_n$ can be vacuous (empty) for all but a finite number of stages (say, for every $n>1$). A reader, or even an author (not me :-), might not even be alerted to the situation. This may happen, as in my notes, that the main case (meaning the only hard case) is the vacuous $B_n$. To me it feels humorous.
My dinner time is approaching. Thus, before I go into something new in my mathematical project (will I ever?), let me mention about daba+dabanese. I am thinking about some metadabagrams, e.g. $\nu$gm (for dabagram), and also about a special qutation instrument, namely a metadabagram $\nu$odb (for no dabanese).
I've created a 2014++ post devoted to daba, namely Daba essay 2014++. It appears in the index of Everything 2014++. And the link to Everything 2014++ is listed back at the top of Daba essay 2014++.
Now I have Number Theory 2014++ file. I'll allow it to start with an index followed by an initial note about prime products. But I write this note not to show of with my embryonal (embryonic) activity but to steal the mathjax script from the header above (and to insert it into the new file--shouldn't I have done once and forever in one place at my ipage site?).
I got a break through (not a big deal except for my own circumstances) last September. Then somehow I stopped working on it (on prime products), I don't know why. This time I got easily and cleanly a better result:
$$\forall_{n\ge 20}\ \,\Pi_\frac n4\ \ge\ n$$
instead of my old
$$\forall_{n\ge 19}\ \,\Pi_{\frac 5{19}\cdot n}\ \ge\ n$$.
Actually, the new result (above) is not perfectly stronger since the constant $19$ in the last formula above is smaller than $20$ above it. The value consoderations are trivial anyway since the results can be easily improved. It just feels good that the proof is so much simpler, free of complications.
Oh, I see why I was getting complications. For instance, in the case of my present Theorem 1 from Number Theory 2014++, in the past, I would aim at constant just under 6 instead of the easier constant 4. My old approach was driven by a consideration like $2\cdot\ 3\cdot 5=30=5*6$. This is why I attempt to get constant nearly 6, and this was somewhat hard. In my present method I decide on the constant first. Then I take whatever initial constant is convenient. Right now I am concentrating at looking at powers of $2$ only, where I consider a smaller integer $m<n$ -- I want $n-m=2^d\cdot k$ where $k$ is an odd integer greater than $1$. At least for the time being (perhaps not for long).
Oh, no! It's not good when my present me does not respect my old me. Last September I was alerted to certain complications, I even handled them in some simple cases but I was not ready to edit my partial results. And now, yes, I have committed an error. Thus my old notes are not really obsolete. During September I considered some simple exponential diophantine equations (if there are such things as simple exponential equations; well, some are, and some hard and pretty general cases were solved). This time I will commit these observations to writing, at least a simple case (I guess that last time I got overwhelmed by the looming task of writing),
This was all true but I also was panicking a bit. A small proof adjustment would keep my Theorem 1 intact. I erased it, which is ook too. Now I'll write it down a lit simpler (I'll consider two simple cases separately instead of combining them in a bit clumsy way).
I am oscillating between being overly optimistic and panicking. It should convergence somewhere sometime, hey! I tried to solve of hand a simple exponential equation. I used to do them several times in the past (not to mention a serious one at one time). I would do the same time after each break in activities. Right now somehow I was like dead. Thus I looked at my September hand made notes, and it was right there done with ease. At the time, September, I got overwhelmed by the magnitude of the possibilities (actually, I had some other issues which have diverted me or simply stifled). Now I may commit more to the 2014++ site. In particular, this time I will open a file for simple exponential diophantine, so I will have them whenever needed.
I finished the needed exponential equation, see Simple exponential diophantine equations; and I applied it to finish Theorem 1 of N.Th. 2014++.