You can set up a Wordpress blog for free. You can also get jupyter notebooks for free which lets you format using mathjax which I think is the same code. When I posted links to the jupyter notebooks before I was using that same code to generate those math symbols.
I can do almost everything with my OpenOffice word processor. The thing I cannot do is get subscripts and supercscripts to line up correctly when I want to use both on a sigma, for instance.Originally Posted by YesNo
What I use are Google docs and sheets for my personal documents. You just need a Google account to get that. It is all in the browser. You can also use mathjax with it after installing a plugin, but I use jupyter notebooks for mathematics with the underlying python kernel so I can calculate right in the notebook. I also use Google to back up all my photos on my phone as well as copy them to my computer (Windows 10).
I was reading more about Lehmer numbers. Any Lehmer number is also a Carmichael number. I can see why Carmichael numbers exist and Lehmer numbers probably don't. The Carmichael function lambda(n) is smaller than the totient, phi(n), and so it has a better chance of dividing n - 1. For example, the Carmichael number, 561 = 3*11*17, each of 2, 10 and 16 divide 560, but not their product.
Right now my head is fpinning in amazement at the fimple divifor functions, which fhow up in all kinds of not-fo-fimple places. They are involved in fome bigtime formulas by powerhoufe mathematicians and even have a clofe connection to the Reimann hypothefis.
We worked out σ1(pk) for ourfelves, and may have gotten to the more general formula if we had kept at it. The fimple functions are pure magic, but one fhould not be amazed at them, for they are there to be understood and are among the more underftandable objects in number theory. The mulitplicity of their connections ftill dazzle. But one can ftare at each one of them and fully underftand why there is a function there. We ftared fo long at σ1 that we know exactly how it works, we have taken the myftery out of it. I have realized I need to ftare now at what Wiki-peja calls σ0. I have not been working or thinking much becaufe I am coming out of a depreffion. Oh, by the way, of courfe one fees many obvious connections of thefe functions to the Euler phi function, which the article explores. The formulas are fuddenly no longer fimple, they look like ftuff Ramanujan himfelf would have worked on or produced in this field, and indeed he and Hardy were working in the immediate area. I will feel much ftronger once I tie up the divifior functions. I am impatient but ftill recovering, for I want to be off to the theory of lower bounds in logarithms. The more myftery I take out of thefe things the better I might feel about it when I have to die.
Last edited by desiresjab; 12-22-2016 at 06:48 AM.
I have forgotten fome of the techniques for calculating limits. I fuppofe reviewing them, then, had ought to be a profitable venture before I look at the theory of lower bounds in logs. I remember there were fome functions you could not tell fimply by looking at whether they converged or not until fome proper manipulation had been done. It fhould take an hour to review what is proper. But when will I get to that hour, being as lazy as I am ambitious yet full of fchemes for learning? I tend to circle thefe propofitions flowly like a dog fizing up its rival, once I have them in my fights. Then I rufh in fnarling.
The proof in this link makes it cryftal clear why there can be a function for the number of divifors of a number. The inconvenience is we ftill have to break it down to prime factorization, the number's pretty face is not enough to give us the number of divifors it has. I knew thefe functions were fimple. Why did I ftay away fo long? Juft lazy or afraid. Cryftal clear. Aren't there only two important ones--the number of divifors and the fum of the divifors? It feems like there was another. Maybe I juft got my notations croffed up.
http://mathschallenge.net/library/nu...er_of_divisors
Lol! Some "f" spam here.
"I seemed to have sensed also from an early age that some of my experiences as a reader would change me more as a person than would many an event in the world where I sat and read. "
Gerald Murnane, Tamarisk Row
It does seem that the "s" becomes an "f" often, but I have had things like that happen with a faulty keyboard. Sometimes it is just my fingers.
I agree that multiplicative functions are convenient until factoring the number becomes difficult. In the case of that divisor count function one would need a full factorization to take advantage of it.
It gives a very nice lisp-like feel to the text!
I thought it might be a joke and reacted accordingly. But I think you are right. It must be a keyboard problem.
"I seemed to have sensed also from an early age that some of my experiences as a reader would change me more as a person than would many an event in the world where I sat and read. "
Gerald Murnane, Tamarisk Row
No, all I am doing is mimicking 17th century writers. Newton and some of the people he communicated with used f's in the places of s's sometimes, but only with rules. I never figured out why they did this. Maybe someone on here knows. I think the rule was at the beginning and in the middle of words but not at the end. It may just be something idosyncratic that developed. It was probably just an elongated s. I don't know the reason. No, folks, I have not flipped, unless you think I already was.
At least I found out who reads the thread. I already thought Danik and DW were.
Now for the simplest connection of all between ф and σ.
ф(n)=σ0(n) where n is a prime. Makes perfect sense, right? The number of divisors, including itself, is equal to the number of numbers less than or equal to it which are not relatively prime to it.
The number of divisors of a prime is just two, itself and 1. Two is the number of elements less than or equal to n which are not relatively prime to it. The two functions are, then, identical in the case of a prime.
I just wanted to throw that in for the viewers. I think it is important and easy to remember, and knowing it might allow you to derive other related functions if there was a need in the wild. Number of divisors and number of numbers not relatively prime to it, are the same thing for a prime. That is where ф will always equal σ0.
By the way σ0 is the divisors of a number. Any divisor is worth 1 in the count of divisors. Whereas σ1 designates the sum of those same divisors.
Last edited by desiresjab; 12-22-2016 at 11:54 PM.
The reason I am dwelling on the σ functions is I now believe they are the most critical functions of traditional number theory. Even the ф function is merely a type of shorthand for σ, it now seems to me. The ф function, which itself is extremely important is really just shorthand for manipulations of the divisor function. That makes the divisor functions incredibly important, when ф is just a special case of them!
Is this vision correct? I believe so, but perhaps I am missing something.
Mathematical formulas in books are indeed succinct and exact, but for that reason all the more difficult to see in their stark simplicty sometimes. Some mathematical propositions that could require great work to unravel without particular insights into this simplicity, are intuitively taken in at a glance with these insights under one's belt.
Let us examine the law that says if two numbers do indeed share a common factor, then that factor will also be a factor of their difference. If A and B share a factor that factor will also divide A-B.
Let us call n the common factor between A and B. Then pn=A and qn=B. But why the general language? Let's get specific. That is what we came to do.
A=6n, B=4n. Now isn't that better? Now there is no problem. Yes, we see easily that the common factor n still divides 6n-4n=2n, the difference of A and B.
When the general language is removed from mathematical propositions, it often serves intuitive clarity. This is the goal. This clarity is exactly what we want. I believe it cannot be possessed at the higher levels until all the elementary propositions are solid as rock in the mathematician's mind. This is why we linger, and not without profit.
Philosophical mood today. Thinking about the history of mathematics. If east injuns 1500 years ago were smart enough to devise and consider what we mistakenly today call Pell's equation (x2-ny2=1), then it would be foolish to believe they did not have possession of propositions in number theory which are more elementary. They did not leap to a proposition like Pell's equation from pure ignorance any more than we would be able to do so today without massive preparation and a solid foundation to build upon. In other words, they had possession of most of what we call elementary number theory.
Perhaps the ancient Injuns and Chinese did not yet have possession of abstract algebra and group theory; perhaps they had no theory of quadratic forms. Then again, we really do not know what some isolated individual might have acheived in his hut, those results being lost forever with his death. By the time European mathematcis began its heyday, Europeans had extreme advantages in communication that had been available to no others before them. Discovered knowledge was cabable of dissemination as never before. What one man discovered, many others had the chance to review and build upon, knowledge no longer had to die in a cave with its discoverer.
We only know of some ancient societies that had this mathematics. We know for certain that many did not. In a pure white landscape it is easy to imagine why Eskimos would not develop even a counting system. North American injuns were mostly not advanced at all, but some of the southern injuns like Mayans and Inca were well advanced, and obviously had possession of some mathematics. This means to me that there were probably some smart Inca holed up somewhere doing number theory for curiosity and pleasure. Sub Saharan Africa seems to have had no advanced cultures. Along the Nile, Tigris and Euphrates there were numerous advanced civilizations. I believe we can assume that one mark of advanced societies is at least the mathematics of measurement and the beginnings of number theory.
Posting Permissions
Reply With Quote