Mmm-1429 In complex numbers there are ten solutions for X: X = 2*exp(2*n*pi*i/10), n = 0,1,2 ... 9 = 2*cos(2*n*pi/10) + 2*i*sin(2*n*pi/10), n = 0,1,2 ... 9 No. Because an equation is not the sort of thing that can pass through given points. If there are finitely many points in the Cartesian plane and you want a (x_1, y_1), (x_2, y_2), ..., (x_n, y_n) and none of the x_i are equal, then y_1 = f(x_1), y_2 = f(x_2), ..., y_n = f(x_n); then look up Lagrange interpolation formula. I prefer pi --Dogstar, a rat's god. :-) Now *that's* clever! (Mmmmmm.... pie.....) ;-) Now THAT is a palindrome! A man, a plan, a caret, a ban, a myriad, a sum, a lac, a liar, a hoop, a pint, a catalpa, a gas, an oil, a bird, a yell, a vat, a caw, a pax, a wag, a tax, a nay, a ram, a cap, a yam, a gay, a tsar, a wall, a car, a luger, a ward, a bin, a woman, a vassal, a wolf, a tuna, a nit, a pall, a fret, a watt, a bay, a daub, a tan, a cab, a datum, a gall, a hat, a fag, a zap, a say, a jaw, a lay, a wet, a gallop, a tug, a trot, a trap, a tram, a torr, a caper, a top, a tonk, a toll, a ball, a fair, a sax, a minim, a tenor, a bass, a passer, a capital, a rut, an amen, a ted, a cabal, a tang, a sun, an ass, a maw, a sag, a jam, a dam, a sub, a salt, an axon, a sail, an ad, a wadi, a radian, a room, a rood, a rip, a tad, a pariah, a revel, a reel, a reed, a pool, a plug, a pin, a peek, a parabola, a dog, a pat, a cud, a nu, a fan, a pal, a rum, a nod, an eta, a lag, an eel, a batik, a mug, a mot, a nap, a maxim, a mood, a leek, a grub, a gob, a gel, a drab, a citadel, a total, a cedar, a tap, a gag, a rat, a manor, a bar, a gal, a cola, a pap, a yaw, a tab, a raj, a gab, a nag, a pagan, a bag, a jar, a bat, a way, a papa, a local, a gar, a baron, a mat, a rag, a gap, a tar, a decal, a tot, a led, a tic, a bard, a leg, a bog, a burg, a keel, a doom, a mix, a map, an atom, a gum, a kit, a baleen, a gala, a ten, a don, a mural, a pan, a faun, a ducat, a pagoda, a lob, a rap, a keep, a nip, a gulp, a loop, a deer, a leer, a lever, a hair, a pad, a tapir, a door, a moor, an aid, a raid, a wad, an alias, an ox, an atlas, a bus, a madam, a jug, a saw, a mass, an anus, a gnat, a lab, a cadet, an em, a natural, a tip, a caress, a pass, a baronet, a minimax, a sari, a fall, a ballot, a knot, a pot, a rep, a carrot, a mart, a part, a tort, a gut, a poll, a gateway, a law, a jay, a sap, a zag, a fat, a hall, a gamut, a dab, a can, a tabu, a day, a batt, a waterfall, a patina, a nut, a flow, a lass, a van, a mow, a nib, a draw, a regular, a call, a war, a stay, a gam, a yap, a cam, a ray, an ax, a tag, a wax, a paw, a cat, a valley, a drib, a lion, a saga, a plat, a catnip, a pooh, a rail, a calamus, a dairyman, a bater, a canal - Panama! -- It is only those who have neither fired a shot nor heard the shrieks and groans of the wounded who cry aloud for blood, more vengeance, more desolation. War is hell. --William Tecumseh Sherman In war, there are no unwounded soldiers. --Jose Narosky The urge to save humanity is almost always a false front for the urge to rule. --H.L. Mencken Some people have no lives.... ;-) (or No it isn't! It was as far as I went. Maybe you were thinking of TAT? Danny Originator: msb@shell.vex.net (Mark Brader) Look, I came here for a good palindrome -- it isn't just the automatic gainsaying of anything the other person says! -- Mark Brader Oh, I'm a programmer and I'm O.K.... Toronto I work all night and I sleep all day msb@vex.net -- Trygve Lode Can anyone out there explain to me how to derive the general form equation of a tangent line, using lnx and ex? It depends on what the line is tangent to. It is usually sufficient to take the 1st derivative of the curve, evaluate at the point of tangency, and then use the standard line equation y - m*(x-x0) + y0 y = m(x - x0) + y0 The general form of a line tangent to a curve y = f(x) at x = a is y - f(a) = f'(a)(x - a) You may want to not blither out your question so quickly, expecting us to mind read your query. What's lnx and ex? Something about a lynx and your ex? Perhaps you mean ln x or ln(x) and exp(x) or e^x. Is that what you mean? Then take the scant time required to assure you're not mumbling. Now even if I've decoded lnx and ex correctly, your question still doesn't make much sense. Do you have a different general from equation of a tangent line than I or are you only wanting the tangent lines for y = ln x and y = e^x. Hey?! Have I finally deciphered your question? Then just plug in f(x) = ln x and f(x) = e^x in the above general form presented at top. What are your results? Maybe this will add insight: Thus, finding the general form equation of a tangent line, using these two functions...... Your notation is confused. Where you write ln_x you mean ln(x) or ln x. Where you write e_x you mean e^x. Where you write f^1 you mean f'. Furthermore, grownups use log for natural logarithm. And the use of a number, say n, to indicate that a function has been differentiated n times has the number in brackets thus: f^(n). f^1 means f. Your confusion is only eclipsed by your arrogance. Keep perpetuating the stereotype. Be sure to stop by my neighborhood, I'll be sure that we grant you the same courtesy. Supposedly at times an idea can be important simply because it shows a surprisingly *easy* way to do something already known where techniques already exist. Now I've talked about what I call my prime counting function before, but here's an explanation for laypeople, and then I want to talk a bit about innovation and the math world. You can count primes several ways. A nice and easy way is to count composites and subtract off their number and 1 to get your count of primes. Like up to 10 the even composites are found just by dividing by 2, discarding any remainder and subtracting 1, like 10/2 = 5, and the 5 evens are 2, 4, 6, 8, and 10, so subtract 1, and you have 4, for the evens 4, 6, 8, 10 and I figured that with the evens out of the way, to find those with 3 as a factor, you should skip evens, so next you have 9 as the only odd with 3 as a factor that's not even. So you have 5 composites, and subtract that from 10 and you get 5, and since 1 is not considered prime, you subtract 1, and have 4, the count of primes, where they are 2, 3, 5, and 7 and just now you just saw an idea fly by which is what makes my prime counting function works, but I'm sure you missed it, so I'll give a bigger example. The even *composites* up to 25 are 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 while the odds that have 3 as a factor are 9, 15, 21 and the one remaining composite has 5 as a factor and it is 25 and the sum of composites is 15, and 25 - 15 = 10, and you subtract 1 for 1, as it's not prime, leaving you with 9 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23 and in case you think I'm just playing silly games with you, let me show you how Legendre's Method does it, as that's an old prime counting method that mathematicians are taught. It uses the same idea for evens, so you have 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 BUT, it take ALL the numbers up to 25 that have 3 as a factor, so you have 6, 9, 12, 15, 18, 21, 24 and because you have numbers in both lists, Legendre's next finds all that have 6 as a factor, so you have 6, 12, 18, 24 and subtracts them from the previous, so now you have 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24 to give your count from the two previous lists. That leaves 5 and Legendre's Method finds all the composites with 5 as a factor, so you have 5, 10, 15, 20, 25 and then it take all the composites with 10, as a factor 10, 20 and all the composites with 15, as a factor, so you take 15 and subtract them out and now adds them back into the other list, so the full list is 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25 which is the same as before, but notice you had this subtracting out and adding back in with Legendre's that wasn't mentioned in my previous. Mathematicians found that Legendre's Method was slow and clunky, hard to work with as when you add more primes you get more and more combinations. So they developed faster methods *from* Legendre's, while one day a couple of years ago I started thinking about prime numbers. You see, I haven't gone to school as a mathematician and had never cared about counting prime numbers before, so I didn't know about Legendre's Method, and when I looked to count primes, I intuitively looked to count the even composites first, and then the odds divisible by 3, and so forth as I demonstrated. I mathematicized that method and you get a couple of formulas. I've given those formulas before when talking about my prime counting function. One is a fully mathematicized version that can find the initial primes on its own, like you don't have to tell it that 3, and 5 are primes as it will figure that out by recursion, and sci.math'ers have routinely belittled that formula for being slow. Now if you give it a list of primes, like how I demonstrated here, where you know that 2, 3 and 5 are primes, like how Legendre's has a list, it is quite fast, and is much faster than Legendre's Method. However, several sci.math'ers have routinely said that my prime counting function is just Legendre's Method. Now then, how can a method that looks short and neat like you have 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 for the even composites, and 9, 15, 21 for the odd composites with 3 as a factor, and 25 for the remaining composite with 5 as a factor compare with a method, which rather naively just gets a count for each prime and then corrects itself, to the extent that for over TWO YEARS some very vocal math people have proclaimed them to be the same method? Well, people want to believe. They want to believe that some guy claiming he has a major math discovery and that mathematicians aren't treating it properly, must be a nut. They want to believe that people they trust don't lie to them. They want to believe it's a sane world. It turns out that I use a very simple and intuitive idea to count primes, which is just so happens is not in any math textbooks that I'm aware of, and I have looked quite a bit. You can go yourself and do a web search on prime counting or prime counting function and see if any links talk about a simple way to count prime numbers which rather sensibly only counts the primes not already counted, versus counting in extras to be subtracted back out. Those of you who think you're brilliant mathematically, derive the formulas that follow from doing the count as I do. I'll admit that I'm puzzled that I get so much arguing about what I see as simple techniques, and even more puzzled that the idea is new to me. To date I've challenged and never seen anyone else capable of doing the simple derivation, which puzzles me still more. I will help you out. The main function I call dS(x,y) where it is the count of composites that have y as a factor that do not have any primes less than y as a factor. For instance, as I showed dS(25, 3) = 3, and those composites are 9, 15, 21 and yes, it may puzzle some of you that you can simply get the answers to that function without calculating, but you can. Notice that if y is not prime then dS(x,y) will equal 0, as then there will be some prime less than y for which each composite will have a factor. Like if y = 4, then, of course, every number that has 4 as a factor will have 2 as a factor, so you see, all the composites with 4 as a factor get counted with dS(x,2). If you're really curious to see the full mathematicized formula, and to see what I want some one of you to derive, check out http://en.wikipedia.org/w/index.php?title=Prime_counting_function&oldid=9142 249 and you will see it on that page as the p(x,y) function. You can also check the current Wikipedia page on the prime counting function to see how others try to explain. Now then, if I have a simple idea I can explain this easily, why do math people mostly call me names? Well, innovation can be threatening. In the real world an innovative idea can prove itself by building better mousetraps, as they say. But in mathematics, mathematicians feel they have the capacity to decide, as a group, what they think is important or not. So, you see, in today's mathematical world, it doesn't matter how brilliant you can show a result to be, even if you can explain it to non-mathematicians, as the professionals believe they own the discipline, and they don't have to answer to anyone--not you, not me, not anyone. So I'm stuck talking about my research a lot on Usenet, where sci.math'ers often make fun of me. You see, they can make fun of me, despite the correctness of my mathematical results, and despite the real value of those results because math society is a true democracy. It's not about the truth in that society, but what the group decides is the truth, as you can see here with a prime counting function, where I found an innovative way to count prime numbers, and math people make fun of me versus cheering that success. James Harris Blichhhh....... Just go to http://mathworld.wolfram.com/LegendresFormula.html Between equation (1) and (2) you find Taking a = pi (sqrt (x)), where pi (n) is the prime counting function, gives phi (x, pi (sqrt (x))) = pi (x) - pi (sqrt (x) + 1... and equation (3) is the recurrence relation phi (x, a) = phi (x, a-1) - phi (x / p (a), a-1) Notice the emotion and the heat of that emotion. 1... However, I can explain by example, and then you can go to that site and see the truth. The sci.math'er who made that reply has lied about my work for over two years now, and his energy in replying as much as he does to keep up his position should tell you something. Here's another example, and then you can go to the link he mentioned and see if there's anything like it. Counting up to 50 this time, my idea first gets the even composites: 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50 and then gets the *odds* that have 3 as a factor: 9, 15, 21, 27, 33, 39, 45 where notice I don't bother with evens that have 3 as a factor, and then you get the odds that don't have 3 as a factor that do have 5: 25, 35 and that leaves composites that have 7 as a factor, but none of the smaller primes, where there is just one: 49 and notice that you have a rapidly shrinking count, from a sensible way to count primes. I just naturally and intuitively came to such a method when I started thinking about counting primes, as it made sense to me. But it's just a basic fact that Legendre's Method does not count that way. The mathematical formulas are somewhat complicated, which is why I now use the simple demonstration of giving out the composites, as I need something simple enough that you can tell when sci.math'ers like Christian Bau are lying to you. If you go to the link he gives, you'll see nothing like the sensible counting method I used described. It's complicated enough how Legendre's works with even such a simple example as the count of composites up to and including 50 that I won't show it in full here, but give you one example, which is how it handles the composites with 3 as a factor. If any sci.math'ers or others who claim that my work is old wish to dispute that by giving a full demonstration with Legendre's, they are free to do so, and *should* if they are trying to be honest. I'll consider how it handles the composites that have 3 as a factor. It takes ALL of them, so you have 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48 and then it finds all the composites that have 6 as a factor 6, 12, 18, 24, 30, 36, 42, 48 and subtracts that number to correct the overcount. For the count of 5 it does that for 10 and 15, and by doing so, it subtracts out 30 *twice* as both 3 and 2 have it as a factor, so there's another correction of an addition of 1. For 7, it has to handle the counts for 2, 3, and 5 in varying combinations, correcting for overcounts, repeatedly. It is dizzyingly complicated to look at for even a simple example. My feeling at this time is that sci.math'ers like Christian Bau and others like David Ullrich, a math professor at Oklahoma State University, got away with a rather basic lie because people would get lost on the equations from the mathematicization. Physicsts can understand that, as electromagnetic theory was worked out by Faraday, and mathematicized by Maxwell, and Faraday couldn't understand the mathematical equations. So I'm going from the mathematicization, where sci.math'ers lie so easily, to the idea so that people can see what the idea really is, and also see that math people are in fact lying to them. And lying quite boldly. James Harris Your ignorance, incompetence, and psychosis are not of interest to the world at large. Quite the contrary. You are not even an interesting laughingstock. Hey stooopid loud troll James Always in error, never in doubt! Harris, put up or shut up. James Harris, King of the Primes! Where are your sceptor and crown, delusional James Harris, your regal clothes? Is a $20,000 prize no questions asked too small to justify your submission of two little prime numbers? Or are you a psychotic impotent gelding? Hey stoopid loud troll James Prime Slut Harris, a better man than you has factored RSA-576. Pookie pookie. http://www.rsasecurity.com/rsalabs/challenges/factoring/faq.html http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html http://www.crank.net/harris.html It's not every braying jackass who earns a whole page at crank.net -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf [snip repost of previous expositions on prime counting] It's customary, when one has already posted information on Usenet, to simply provide a link to previous posts. It isn't really necessary to persist in starting new threads with old information. Besides, you recently claimed you were going to reduce the number of threads you start -- this one is completely unnecessary. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com [snip crap] Your ignorance, incompetence, and psychosis are not of interest to the world at large. Quite the contrary. You are not even an interesting laughingstock. Hey stooopid loud troll James Always in error, never in doubt! Harris, put up or shut up. James Harris, King of the Primes! Where are your sceptor and crown, delusional James Harris, your regal clothes? Is a $20,000 prize no questions asked too small to justify your submission of two little prime numbers? Or are you a psychotic impotent gelding? Hey stoopid loud troll James Prime Slut Harris, a better man than you has factored RSA-576. Pookie pookie. http://www.rsasecurity.com/rsalabs/challenges/factoring/faq.html http://www.rsasecurity.com/rsalabs/challenges/factoring/numbers.html http://www.crank.net/harris.html It's not every braying jackass who earns a whole page at crank.net -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf Yeah, democracy sucks. That's why this country is a Constitutional Republic and not a democracy. is I bet you wish there was some form of Constitution for Mathematics to protect your truth from the tyranny of democracy. Think about that the next time you want to advocate the overthrow of the Constitution by foreign powers. ! AHHHA 7x multi-palindromes!most brilliant! AHHHA palindrome! solve, multiplication sum. (AA x AHA x AHHHA )^2 = ADMIT ONE NOT IMDA. (AA x AHA x AHHHA )^2 = ADMIT ONE NOT IMDA. very easy + beautiful.. each letter is replaced by a different decimal digit, 0-9. the equation is true. but nobody can solve a simple puzzle. i promised a satisfying reward. it even works on telephone keypad as well, nearly. also nearly binary ! SPOILER SPOILER SPOILER SPOILER SPOILER 11.3.05 ... the answer has been previously posted. AN 8-DIGIT PALINDROME.AN 8-DIGIT PALINDROME. more below... Set of puzzles, below. Submission for your magazine. 11.08.03 00:08 08.04.04 22:53 sent. 19.12.04 22:11 ..edited expanded... more. SENT. Don.McDONALD. There is no such thing as a multiplication sum. Multiplying is multiplying, summing is summing. A=D=E=H=I=M=N=O=T=0 Solved. Reward? Right side < 10^15 (10 * 100 * 10,000)^2 = (10 * 10^2 * 10^4)^2 = 10^14 Now if a = 2 instead of 1, then (2 * 2 * 2)^2 * 10^14 = 64 * 10^14 = 6.4 * 10^15. Thus a = 1 (11 * 1h1 * 1hhh1)^2 = 1dm ito nen oti md1 and right side < 2 * 10^14 If h = 2, then left side = 265 * 10^12 = 2.65 * 10^14 Thus h = 0, and left side = (11 * 101 * 10001)^2 = 11,111,111^2 = 123 456 787 654 321 Thwe number one claim of a fraud. It is all a conspiracy against the truth. Why? Surrogate stupidity is what I call people like you who try to monopolize the market on how much stupid one person can fit into one head. Okay, you win! -- Oppie the Bear aka TOJ (The Other John) 'Remove' MYWORRIES to email me! If life gives you lemons, squeeze the juice into a watergun and shoot other people in the eyes.