I'm just kinda curious as to what people think regarding the solution to the Riemann Hypothesis. For those who don't know, the Riemann Hypothesis states that all of the non-trivial zeros of the Riemann-Zeta function (which has a ton of badass properties) have real part 1/2. This problem is generally considered the most important, difficult, and famous unsolved problem. Solving it would make you amazingly famous and probably rich. If you don't know about the Riemann-Zeta function, I highly suggest reading about it. It was first proposed by Berhard Riemann in 1859 and has resisted attack since that year, although a lot of "progress" has been made on it. I find it to be the most fascinating function in the world of mathematics, especially regarding its relation to the distribtion of prime numbers. Pimp. So what do you think?
Thanks. I am weak on math, but will follow the post here. Mainly want to encourage others to look by removing the the zero from the "number of replies" I do have two slightly related questions: (1)I have the vague impression that Fermant's last theorm has finally been proved. This impression is confirmed in part by your comment: "most important, difficult, and famous unsolved problem." Can you re state it for us (I recall it was easy to state and understand, but forget it.) and tell what the result was (i.e. is it true or false.) (2) Was it Gauss who, as approximately 12 year old, proved that if the three pairs of opposite sides of an irregular hexagon which is inscribed in a circle are extended, they intersect in three points that all fall on the same straight line? I consider that kid, one of my "heros" but am embarsed to say I have forgotten that mathetician's name. (I tried to prove this geometrically for a few hours before giving up and drawing a few examples to convence myself that it is true.)
Yeah, you have the right impression about Fermat's Last Theorem. Fermat's Last Theorem (not to be mistaken with Fermat's Little Theorem, which deals with congruences), states that there are no whole number solutions to the equation a^n + b^n = c^n for n>2. Obviously there are an infinite number of solutions for n=2 (Pythagorean Triples) and trivially for n=1. I believe that Fermat originally stated in the margins of a book (in Latin, perhaps?) that no cube can be written as the sum of two smaller cubes, no fourth power as the sum of two smaller fourth powers, etc. He also claimed to have an "elementary" proof for this, but that it was too long to include in the margins. Andrew Wiles pretty much locked himself in his house for something like 7 years before he finally proved the theorem true using modular forms and elliptical curves, although from what I've read of the proof he didn't really prove a whole lot of new stuff himself; rather he tied together a bunch of other mathematicians' work to arrive at the final result. As far as Gauss goes, he certainly was one hell of a mathematician. I'd have to say he's tied with Euler as my favorite of all time. I don't know about the hexagon thing, but Gauss was the one who, in first grade, added up the numbers from 1 to 100 in minutes when his teacher gave the class that exercise to burn a little time. Even more amazingly, he pretty much conjectured the Prime Number Theorem straight up when he was 15, saying that the Prime Counting Function is asymptotal to n / [ln n], which he later improved to the log integral. What a badass. Hope this helps.
http://math.cofc.edu/faculty/kasman/MATHFICT/mfview.php?callnumber=mf457 i liked the public key encryption part, did he invent that too?
Thanks for the link, leopold. I was psyched when Riemann came up in that episode. I'm not sure, but I doubt Riemann came up with encrypting data using prime numbers. It seems to me that that sort of thing wouldn't come up until the advent of digital information, which was long after Riemann's time.
Here's a link explaining a lot about the history of public key cryptography, so I don't have to type it out from memory Please Register or Log in to view the hidden image! http://en.wikipedia.org/wiki/Public_key_cryptography As for the Riemann Hypothesis, yes, I believe it will be shown in the next hundred years, validating the rather large amount of provisional math that has been done with it as an assumption. It is a highly interesting subject, and it has deep structural connections with number theory, which at first seem very strange and almost like magic (kinda like complex analysis, when you find out that you can determine the value of a class of complex integrals by evaluating the encircled poles of the function at hand). But progress is constantly being made in deciphering its secrets, and so I believe that someone will (comparatively) soon put it all together, and provide a solid proof. I, personally, think that the greatest unsolved problem is the question of whether P = NP. Unlike the Riemann hypothesis, this problem has not given us anything to suggest whether or not it is true or false (or independent of the axioms, like the Continuum Hypothesis), other than that noone has given a counterexample (a polynomial time decision algorithm for a provably NP problem) which is circumstantial at best. Even Fermat's Last Theorem had a lot of inroad in the forms of reducibility to another problem. P vs. NP hasn't got anything like that. This problem, IMO, will have to wait for a truly stunning insight a la Newton, to be solved. A funny thing I just realized: If P = NP is undecidable, then it automatically true, although not provable. Why? Because if it undecidable, then no P algorithm for any NP problem can exist, because that would make P = NP decidable by counterexample.
I didn't think this was Gauss. I think his most famous geometric teenaged result was proving a 17 sided regular polygon is constructable with straightedge and compass (He later completely characterized all constructible regular n-gons). Google quickly turned up Pascal for the hexagon thing you're after: http://mathworld.wolfram.com/PascalsTheorem.html It's impossible to tell if the Riemann hypothesis will be solved one way or the other in the next 100 years, or if we ever will. While there are many significant results, they are still quite far from the whole thing. Riches do await in the form of the $1 million Clay prize, but only if it's proven to be true. If someone proves it false, say by a counter example via the many extensive computations going on, no clay prize is to be had. Eitherway (true or false) it's resolution would put your name in the history books forever. It's difficulty has been greatly underestimated in the past, Littlewoods doctoral supervisor assigned it to him as a thesis problem. Hilbert once compared three problems, expecting the Riemann Hypothesis to be solved in a few years, Fermat's last theorem in his lifetime, and that 2 to the power square root of 2 is transcendental possibly never. In hindsight, he had ithe relative difficulties completely backwards. Riemann didn't come up with anything with crypto applications in mind. As far as I know the first use of primes in cryptography is the RSA algorithm from some 30 years ago, and number theory before that had no cryptography goals. Riemann published only one paper on number theory, certainly one of the most important. It's a scant 8 pages to boot, "On the Number of Primes less than a Given Magnitude". You can find translations online, or in the back of H.M. Edwards nice book "Riemann's Zeta Function".
From now on, let it be known to everyone...that I love you Please Register or Log in to view the hidden image! Haha..are you a math major? If so, even if not, teach me something...I love math, and would like to learn more. Lol.
Beavis, I am a Computational and Applied Mathematics / Mathematical Economic Analysis double major. Quite the mouthful, huh? What do you want to know about math? If you just want to learn new stuff, I would recommend reading about prime numbers. They are the most fascinating aspect of math that I have found thusfar, hence the post about the Riemann Hypothesis. What does the Riemann Hypothesis have to do with prime numbers? I suggest reading "Prime Obsession" by John Derbyshire. The beginning is pretty elementary, but the second half will blow your mind. I've read the book 4 times over and am still amazed by it all. And www.mathworld.com is a good place to read about pretty much absolutely anything you could ever be interested in mathwise. Good luck, Beavis.
If you really want to be amazed, ditch Prime Obsession and hit the theory full on. Derbyshire has to make some sacrifices for his intended audience so you miss out on some fundamental and very nifty results. I'd suggest edward's book that I mentioned above for a few reasons. It is essentially devoted to explaining the details of Riemann's paper, so it mostly follows the 'classical' progression of results. This is the way derbyshire proceeded, he appears to have largely used edwards as his source for the math, so it should be familiar (though edwards is much more thorough, it's a full fledged textbook afterall). It follows these classical lines as it's essentially devoted to filling in the gaps of Riemann's monumental, must-read number theory paper (which is included in edwards). If you still hunger for more, Titchmarsh's "The Theory of the Riemann Zeta-function" is really the fundamental book to read. Even though it's old, it's still referenced in just about every analytic number theory paper I've read. He doesn't spend much time on prime number theory, concentrating on the zeta-function itself, so a nice companion is Ingham's "The Distribution of Prime Numbers", which is still a classic. These books are much heavier than edwards though, infinitely heavier than derbyshire, but also far more rewarding from a mathematics standpoint. Given your computational background, you might especially enjoy the fairly extensive chapters in Edward's on how to locate zeros of zeta (there's a bit in titchmarsh as well). Though older, they still contain much of the basis of modern computation as I understand it (which suprisingly goes back to Riemann himself, I think Derbyshire mentioned the Riemann-Siegel formula). If you want the latest algorithms used, Andrew Odlyzko's webpage is the place to start.
Well Gauss did write down something pretty much similar to his later prime number conjecture when he was 14 or 15. This wasn't "straight up" as he obviously spent hours looking through massive tables of prime numbers. In the 18th and 19th centuries, before digital computers many people were employed to make tables, astronomical tables, logarithmic tables, trigonometric tables, prime number tables and so on. These guys were called computers, since that is what they did, they computed. All sorts of conjectures were floating around as to the distribution of the primes. It would be quite odd if these computers never even considered if there was any trends or patterns in their tables. Anyway long story short, Gauss' conjecture turned out to be the most accurate, along with Legendre's which was pretty similar. It was really only when Chebyshev decided to investigate the PNT in the 1850s that Gauss' conjecture became assailable. CHebyshev proved (among other things) that if the limit as x -> infinity of pi(x)ln x/x exists then its value is one. I.e if the limit exists pi(x) ~x/ln x as Gauss conjectured. Hadamard and de Vallee Poussin both proved that this limit does indeed exist, based on work by von Mangoldt. It was known that the PNT, Gauss' conjecture, was equivalent to Psi(x) ~ x, where Psi(x) = x -SUM(X^r/r) + SUM(x^(-2n)/2n) + k (This is not how Psi(x) is defined but Von Mangoldt derived this result in the 1890s). Where k is just a constant, the first sum is over r, the non-trivial roots of zeta(x) (considering x a complex number) and the second sum is over the positive integers, n. So if Psi(x) ~ x then x -SUM(X^r/r) + SUM(x^(-2n)/2n) + k ~ x which boils down to the more simple result that lim x-> infinity SUM(X^(r-1)/r) = 0. To prove this result Hadamard and De La Vallee Poussin did some heavy duty number theory and complex analysis. As someone has already said H. M. Edwards' book is pretty decent at explaining all this, but you should have some knowledge of complex analysis before reading otherwise you won't get through the first chapter. Also, the book was first published in 1974 and I'm not sure how things were taught back then but he seems to have an odd way of doing contour integrals, which kind of bogs down the book a bit.
Nice post, I just want to add a couple things. The usual prime counting function, pi(x), is just counting primes less than x. Those who have read derbyshire will have met a function called J(x), which counts prime powers p^n with weight 1/n, and you could relate them to one another (to get pi(x) in terms of J(x), Mobius inversion was used). This Psi(x) beast mentioned above counts prime powers, p^n, but with that can be thought of as (1/n)*log(p^n), or just log(p). It's relatively simple to get an integral involving Zeta(s) that gives J(x) or Psi(x) exactly, the hard part is evaluating this integral in a useful way (like what von Mangoldt did for Psi(x)). It turns out that Psi(x) is easier to work with, and this is what Hadamard, de la Valee Poussin and nearly every modern treatment does. This integral that gives Psi(x) involves Zeta'(s)/Zeta(s)*x^s/s over a certain contour (that Zeta' is the derivative of Zeta). In the formula given by von Mangoldt, both the infinite sums are over the zeros of Zeta, SUM(X^r/r) over the non-trivial zeros as mentioned, SUM(x^(-2n)/2n) over the trivial zeros, -2, -4, -6, etc. This is a result of the residue theorem from complex analysis which lets you express contour integrals in terms of the poles, essentially the points where the integrand "explodes" which is what the 1/Zeta(s) term is doing at the zeros of Zeta. (Incidently, the main term "x" comes from the pole of Zeta' at s=1, the constant from the pole of 1/s at s=0). Hadamard and de la Valee Poussin didn't quite prove directly that lim x-> infinity SUM(X^(r-1)/r) = 0, they used slightly different versions of von mangoldt's explicit formula that was easier to work with. The important step in any case was showing zeta had no zeros on the line real part s=1, then the individual terms themselves of this sum will tend to zero as X->infinity (justifying a term-by-term limit like this is necessary, this is what their variants made simpler). About Edwards, a basic understanding of complex analysis is absolutely essential. I do think it's one of the easier introductions to zeta available these days. The main oddity with his contour integrals that I can remember is when he writes an integral with lower limit "+infinity" and upper limit also "+infinity" for a contour that involves the positive real axis twice in two different directions. This is really odd notation, most authors would describe the contour, call it C, and use C in the integral notation.
Thanks, I hereby officially designate him as my hero. He was very good in physics arguments also. I have reads some of the letters he exchanged with some unfortunate Bishop whose sad lot was to try to get Pascal to admit that "nature abhors a vacuum" was why barometers work etc. If I had to defend fact that 2+2 = 4 and Pascal was arguing that it did not, I would just quickly conceed, rather than try to match brains with him. Having also read most of Newton's great book, and been very impressed with geometry as a proving tool, I would like to know if I am still correct when I tell others that: In Newton's day, geometry was the most accepted, if not the only accepted, form of "proof," just as today some mathematicians do not today accept "exhaustive computer demonstrations" as "proof." Based on your comments above, perhaps they (computer proofs of postulate truth*) are now considered "proofs" - any comments? ______________________ * Clearly a counter example would be accepted as disproof, even if the finder "cheated" and used a computer to find it.
I was specifically refering to the possibility of finding a counter example to RH by a computer. The actual math behind this kind of work is very much understood and relatively simple. Riemann himself appears to have proven the first few zeros (ordered by imaginary part) were on the critical line by some hand calculations. Confirming they are all on the critical line up to some given height is just more of the same (though algorithms have improved slightly), but gets beyond the range of human computation very fast. If someone claimed a zero off the line, it would be checked by independants very quickly, and would quickly be accepted by mathematicians. It's not at all an impenetrable process and this kind of computation is generally accepted as much as you would trust a calculator to add 2 and 3 (which is to say much work still goes into verifying your program). I should stress that no amount of zero locating by computations like this can prove RH. You will only ever have a finite number of zeros this way, not even a drop in the bucket (there are infinitely many non-trivial zeros). It does make it look more and more likely though You might want to look into the histroy of the Four Colour Problem (google it). It was initally proven in the 70's via some very extensive computer work and was met with much skepticism.
I read about that. The mathematicians working on it first had to show that all graphs reduce to around 2000 special cases, then they used the computer to verify those. I think afterwards another fellow found a simpler computer proof.
It's a good website for getting definitions, but I find it difficult to navigate. That is, if I don't understand the concepts assumed on one page I'm not sure where to start.
