Liczby pierwsze. Liczba pierwsza to liczba naturalna, mająca dokładnie dwa podzielniki: dzieli się przez 1 oraz przez samą siebie (liczba 1 nie zalicza się do liczb pierwszych, gdyż ma tylko 1 podzielnik). Liczby naturalne większe od 1, nie będące liczbami pierwszymi to liczby złożone; można je rozłożyć na czynniki będące liczbami pierwszymi. Początkowe liczby pierwsze to: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, ... Stąd można ściągnąć skompresowany plik tekstowy zawierający 78.498 początkowych liczb pierwszych ( z zakresu do 1.000.000 ). Liczb pierwszych jest nieskończenie wiele. Udowodnił to już ponad dwa tysiące lat temu Euklides.
Załóżmy, że zbiór liczb pierwszych jest ograniczony. Gęstość rozmieszczenia liczb pierwszych wśród liczb naturalnych maleje wraz ze wzrostem tych liczb. Wykres poniżej (sporządzony wg tabeli, której fragment widać obok) przedstawia procentowy udział liczb pierwszych wśród liczb naturalnych od 1 do 20.000.000. x = lnN czyli: N = ex dN = ex . dx k = 5. Jan K. PatrickJMT.
"The Best of edw519" is now free. Reverse Happy Birthday! - edw519. The Best of edw519 A Hacker News Top Contributor by Ed Weissman Copyright 2011 by Ed Weissman. All rights reserved. Foreword Who am I? Chapter 1 - Advice to Young Programmers 1. Chapter 2 - Education 21. Chapter 3 - Careers 31. Chapter 4 - Work Habits 49. Chapter 5 - The Programmer's Lifestyle 87. Chapter 6 - Philosophy 119. Chapter 7 - Building Stuff 158. Chapter 8 - Software Business 187. Chapter 9 - Enterprise Life 210.
Chapter 10 - Selling 232. Chapter 11 - Just for Fun 245. Who am I? My name is Ed Weissman and I've been programming professionally for 32 years. I've done work for many companies, both enterprises and small/medium businesses. I started out on IBM mainframes, moved to mini-computers, then to PCs, and finally to web-based technologies. I've started three businesses, two with partners and one alone, selling both services and products. I've worked with hundreds of people on over a thousand projects and encountered over a million lines of code. I never get too technical. (Thanks, Mom.) 1. 2. Akademia. Understanding the Fourier transform » #AltDevBlogADay. Yes, I realize that after reading the title of this post, 99% of potential readers just kept scrolling.
So to the few of you who clicked on it, welcome! Don’t worry, this won’t take long. A very long time ago, I was curious how to detect the strength of the bass and treble in music, in order to synchronize some graphical effects. I had no idea how to do such a thing, so I tried to figure it out, but I didn’t get very far. Eventually I learned that I needed something called a Fourier transform, so I took a trip to the library and looked it up (which is what we had to do back in those days). What I found was the Discrete Fourier Transform (DFT), which looks like this: This formula, as anyone can see, makes no sense at all. Eventually, I was able to visualize how it works, which was a bit of a lightbulb for me.
Disclaimer: my math skills are pitch-patch at best, and this is just intended to be an informal article, so please don’t expect a rigorous treatment. The Usefulness of Useless Knowledge. By Maria Popova “The real enemy is the man who tries to mold the human spirit so that it will not dare to spread its wings.” In an age obsessed with practicality, productivity, and efficiency, I frequently worry that we are leaving little room for abstract knowledge and for the kind of curiosity that invites just enough serendipity to allow for the discovery of ideas we didn’t know we were interested in until we are, ideas that we may later transform into new combinations with applications both practical and metaphysical. This concern, it turns out, is hardly new. We hear it said with tiresome iteration that ours is a materialistic age, the main concern of which should be the wider distribution of material goods and worldly opportunities.
Mr. Flexner goes on to contend that the work of Hertz and Maxwell is exemplary of the motives underpinning all instances of monumental scientific discovery, bringing to mind Richard Feynman’s timeless wisdom. This lament, alas, is timelier than ever.