taeto

Nor is your patience infinite.

Since several passengers on that plane were Canadians, maybe to be openly joking in public is not a great idea now, nor for a good while yet.

No sweat. I also do not know whether a quantum regime is scientifically important or a meme, or perhaps something in between. Is there no time duration involved? Maybe the original poster will explain it better.

It was motlan who used it. I explained it already.

If you can look up "meme" on wikipedia, maybe you can also look up "quantum realm"?

The young kids call it "meme".

Thank you for a very nice answer! I did not really want to target those who post in the speculations section particularly. Just found it convenient to use that same word. Come to think of it, crackpots in a broad sense seem to include those who see scientific progress as a bad thing. Like in mathematics, cranks will see it as a bad thing that Cantor discovered that there are infinite sets of different cardinalities, and that Gödel and Turing discovered true statements that are not provable and computable. And of course in physics that Einstein described theories that are similarly hard to understand based on everyday intuition, like black holes. In other sciences, Watson and Crick were severely criticized as well. Maybe a crackpot is the one who attempts to "disprove by argument" established science because it is new and seems scary, if you are uneducated. And a "speculator" is one who simply worries about scientific observations, just on the basis that they are new and seem scary, if you are uneducated. Clearly not everyone who submits to the speculations section belongs in the latter category, just as not everyone who submits in a science section without understanding everything belongs in the crackpot category.

Distance scale \(10^{-9}\) m, and near-zero temperature.

How about a "speculations posting" bingo? The first square could be "Will this new discovery kill us during our lifetime?" Don't know how I came up with that particular example exactly 😆

Usually it is called \(\varepsilon.\) Whatever it is called, I suppose it is a positive number. In which case it is correct what Ghideon points out.

Is this not an engineering topic?

In conclusion, if we replace your \(8\sqrt{2}\) by \(4,\) then the two functions agree whenever \(|\phi-\phi_0| < \pi,\) where difference is taken modulo \(\pi,\) and otherwise they do not. If you replace \(8\sqrt{2}\) by \(4,\) and replace the LHS by its absolute value, then the functions agree everywhere they are defined.

Then \[4\cos\theta (1-\sin \theta)\sqrt{1-\sin \theta} = [1+\cos (2\theta)]^2 = 4\cos^4 \theta\] becomes equivalent to \(\cos \theta = \sqrt{1-\sin \theta},\) and by Pythagoras's \(\cos^2 + \sin^2 = 1,\) this is equivalent to \(-\pi/2 \leq \theta \leq \pi/2\) if we assume \(|\theta|\leq \pi.\) That is since if \(\theta\) lies outside this interval, \(\cos \theta\) is negative, whereas \(\sqrt{1-\sin \theta}\) is non-negative.

When I copy/paste the first equation and enclose it between \ [ and \ ] (without the blanks) I get \[\frac{1}{8\sqrt{2}\cos{\frac{\phi-\phi_0}{2}}(1-\sin{\frac{\phi-\phi_0}{2}})\sqrt{1-\sin{\frac{\phi-\phi_0}{2}}}} = \frac{1}{[1+\cos{(\phi-\phi_0)}]^2}.\] Neither expression is defined for \(\phi = \phi_0 +\pi.\) For all other values of \(\phi\) the equation is equivalent to \[8\sqrt{2}\cos\theta (1-\sin \theta)\sqrt{1-\sin \theta} = [1+\cos (2\theta)]^2,\] where \(\theta = \frac{\phi-\phi_0}{2}.\) For \(\theta = 0\) this is not right, since \(8\sqrt{2} \neq 4.\) Did you copy your equation correctly? You say that your graphs coincide, so they must show something different from this. Or just that they "have the same shape", but one might be a multiple of the other? We can try to use \(\cos (2\theta) = 2\cos^2 \theta - 1,\) which makes \([1+\cos (2\theta)]^2 = 4\cos^4 \theta.\)

Typically they are boiled in water or oven baked at low heat. They are treated and have taste much like shrimp.

It is clear that you did a lot of work just on the \(10\times 10\) case. Maybe you did much less work on the \(n\times n\) cases for \(n < 10\) or \(n > 10\). What is your feeling about what happens if you consider one of those other cases? Say, for \(n = 20,\) maybe the typical relationship will no longer be 3:2 or 2:3, but maybe 5:2 or 2:5. Will you continue to study these more general cases, or will you encourage somebody else to continue your studies?

You choose \(10\times 10\) because the results are more significant than for other sizes? Would you be able to explain exactly how the results for, say, \(7\times 7\) matrices are not as significant as for \(10\times 10\) matrices? So we can at least understand what you mean by a "result" and how to rate its significance.

I recognize some of these from recent posts . How large is an actual bingo card? I suspect you soon have to get busy about weeding out entries to make room for new ones going in.

You answer "so-called" means "thus-called". Which aligns with what Strange said. But your text would change into having a paragraph starting: Until now, the definition of thus-called prime numbers did not allow etc. So then this newly added "thus" does not refer to anything previous. This language problem persists, and you have to revise your text for certain parts of it to make sense. And did you think about \(15\times 15\) matrices? And do you actually not think that the corresponding \(5\times 5\) matrices would make a good introduction for beginners too?

Your definition is fine. It defines exactly what you want it to define: what we get by adding 0 and 1 to the primes. But you lie when you state that the whole numbers are not divided into two disjoint classes, primes and non-primes (the expressions themselves actually say that much). This sounds like it is a motivation for being interested in your new study object, that you claim has a superior property, does it not? Therefore as soon as we see that you are wrong, which is very soon, we should have to lose interest. How is that a good thing for you, do you think?

I have eaten grasshoppers when visiting Oaxaca; indeed, garlic, lime and salt make up the standard accompaniment. I would hold back on the butter, it is already heavy food. Another tradition is to serve them on fried eggs. If you are too squeamish, pass them through the digestive system of the chickens first.

Don't worry, there will come lots more views in the future. I have seen it 🥴.

For those interested in "coincidences" of values of physical constants and their history, there is a well-written (don't let the typesetting fool you) paper by Victor J. Stenger: https://web.archive.org/web/20120716192004/http://www.colorado.edu/philosophy/vstenger/Cosmo/FineTune.pdf Jean-Yves, I have read your article, and the amount of work that you put into it is obviously impressive. So far as I can see it presents only the consequences of your concept of ultimate numbers up until the number 99, is that not essentially correct? Can you make any predictions, or have you even made observations already, concerning the consequences up to as high as 224 (so that you have square \(15\times 15\) matrices available)? I find confusing sounding quotes such as (...) the definition of so-called prime numbers did not allow the numbers zero (0) and one (1) to be included in this set of primes. Thus, the set of whole numbers was scattered in four entities: prime numbers, non-prime numbers, but also ambiguous numbers zero and one at exotic arithmetic characteristics. You are aware that there is a precise definition of primes, according to which every integer less than 2 is a non-prime, including 0 and 1? So why explain the opposite in your article? And why so-called prime numbers? I do not know how it works in French, but you should realize that this sounds insulting. You should remove such phrases from your paper if you want to be taken seriously.

I would not mind looking at your paper. But please tell me more about which media and audience you intend it for. Not for a dedicated number theory journal, I presume? You might write a monograph?! This comment is confusing. I got convinced from reading your initial post that the sequence of ultimate numbers is exactly the same as the sequence of the primes with 0 and 1 appended at the beginning. How can that description be ambiguous? If you are only saying that "0, 1 and the primes" is not "a name", then maybe you have a point. But would you expect them to write "this is the sequence of ultimate numbers, where an ultimate number is defined as a prime or one of 0 and 1"? It doesn't seem natural.

Sensei, you are right of course. And this assumes the old-fashioned classification. In this new-fashioned classification, the biprime 4 is in the raised class, and the biprime 6 is in the (pure) composite class.