Possible Proof of ABC Conjecture 102
submeta writes "Shinichi Mochizuki of Kyoto University has released a paper which claims to prove the decades-old ABC conjecture, which involves the relationship between prime numbers, addition, and multiplication. His solution involves thinking of numbers not as members of sets (the standard interpretation), but instead as objects which exist in 'new, conceptual universes.' As one would expect, the proof is extremely dense and difficult to understand, even for experts in the field, so it may take a while to verify. However, Mochizuki has a strong reputation, so this is likely to get attention. Proof of the conjecture could potentially lead to a revolution in number theory, including a greatly simplified proof of Fermat's Last Theorem."
Re:Rarely much smaller than? (Score:5, Informative)
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Obviously, that's the preliminary intuitive statement. Look further down the page for the formal statement.
Re:Rarely much smaller than? (Score:5, Informative)
That is precisely the point of the proof, to determine under which conditions the sum of 2 integers is less than the product of the prime divisors of the 3 original numbers. I hope that is less vague :P
Re:Rarely much smaller than? (Score:4, Informative)
"Rarely much smaller than"? What kind of mathematical statement is that? Are we to assume that most of the time, d is somewhat smaller than c? Are there conditions where d is larger than c? How are you supposed to get anything done with vague statements like "rarely much smaller than"?
There exists mathematical statements which sounds rather "unmathematical" at first, as an example, "almost everywhere" has a precise meaning in measure theory.
http://en.wikipedia.org/wiki/Almost_everywhere
Won't really make for a simpler proof... (Score:2)
Assuming the paper is correct and as impenetrable as the summary claims, this won't simplify the proof of FLT. It'd be a massive rug that the hard parts of of FLT would be swept under.
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A strong form of the abc conjecture (one providing an actual bound, not just showing there is a bound) combined with existing, relatively straightforward, proofs of the truth of FLT for small exponents would indeed prove FLT in general. However, I haven't heard anyone suggest just yet that an effective bound can be obtained from Mochizuki's work. At this early stage, certainly no one but Mochizuki would know, if even he does.
Re:Linking to Wikipedia to explain math (Score:5, Insightful)
Nobody's measuring anyone's penis--the truth is a lot more boring (and reasonable) than that. Wikipedia is a fantastic first reference for working mathematicians or grad students--I'm sure nearly all math article editors are in these groups--who just want to quickly find out e.g. what the hell an "ultrafilter" is. And so the articles are written in a way that makes them most useful to the people who donate their time to produce them. It's not that any (non-douchebag) mathematician gets off on throwing around smart-sounding jargon. It's just that you can't actually do anything with "intuitive" descriptions.
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No, you can't actually learn abstract mathematical ideas by basing your understanding on intuitive descriptions. If you think you have learned a concept that way, I can almost guarantee that your understanding is faulty. (I've learned this the hard way: I happen to be a mathematician who is particularly adept at providing comfortable metaphors that cause non-mathematicians to believe they've understood something when they really haven't.)
For anyone with a suitable background (say, a first-year graduate st
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For anyone with a suitable background ..., Wikipedia's math articles are generally the best, most accurate and most comprehensive free source of basic mathematics information available. If you don't have that background, no article of any kind is going to be explain to what a "scheme" is, for example. To think so is as naive as believing that you can understand all the nuances of Baudelaire's poetry without learning French; you may think you learn something from a translation into your language, but you actually don't.
Goethe's comment is relevant here:
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and it immediately becomes something entirely different.
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Corollary: Any time a foreigner tries to speak to a Frenchman in his own language, the Frenchman immediately takes it as an insult and an invitation to heap scorn on the foreigner.
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Maybe you can't "learn abstract mathematical ideas" from intuitive descriptions, but 99% of the public don't need to explore all the implications of mathematical ideas. Frankly, the attitude that "everybody should know as much of my art as I do" is quite elitist.
Intuitive descriptions will help the rest of us to a) communicate with the expert who actually understand the implications and b) apply them to real life problems without the need to have a whole understanding. People consulting an encyclopedia don'
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The problem is when the "links right in the intro" form a loop and none of the articles gives a sensible explanation of the field, only unintuitive formal definitions with no practical application.
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Lot's of professional mathematicians. If you sit in the back row of a conference a third of the laptop screens have wikipedia open to get a good quick first idea. Often enough so to be able to work with it afterwards.
Re:Linking to Wikipedia to explain math (Score:4, Informative)
Which makes it even more non-sensical to post it here, on slashdot, a general-interest geek site, where only very few are working mathemeticians or grad students.
A page like this: http://abcathome.com/conjecture.php [abcathome.com] would have been more apropos. No reaching for the jargon, and an actual mini-tutorial on what an ABC triple is and what the conjecture is.
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BMO
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Well WP math articles aren't designed so that every concept comes with a layman's introduction; that would involve massive duplication and bloat. And so, yes, the link you posted would be more appropriate here than a WP link. But I really don't see how you get from there to accusing the volunteer WP math editors of having a big willy contest. There's a reason those articles are written the way they are, and it's not just to make you personally feel stupid. They don't give a shit how smart you think they
Re:Linking to Wikipedia to explain math (Score:4, Interesting)
Because if you look in all other commercial encyclopedias (encyclopediae?), you get a more english (well, natural language) translation of the concepts of a math article. But not even that, Wikipedia on this subject fails even at the post-secondary textbook level. I don't count myself among the dumbest of the population, but when I go to a Wikipedia page for something that is on my level for math, the articles on things like cycloids and such are much better explained by Machinery's Handbook or any other source, really, than there.
I am not saying that Wikipedia should dumb its articles down to the point where even the most innumerate among us would understand all of them, but the "spam equations on the wall with little explanation" model doesn't work very well unless you are immersed in the subject. For example, concepts covered in Algebra I and II in high school should be written for that level.^1 Also, this "write for the grad-student and mathemetician for everything" model does little to help people who use applied mathematics. Indeed, this whole focus on grad-student and up writing in the math articles is at odds with the rest of the Wikipedia.
As a result, anyone wishing to *learn* anything about math is better off using anything but Wikipedia.
Your response to me that the articles are written by grad students and mathemeticians (not all mathemeticians are jerks, btw) for grad students and mathemeticians reinforces the fact that it certainly seems like a giant circle jerk.
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BMO
Footnotes:
1. I had to explain to a high school student that she should not be using Wikipedia for help in her Algebra II class. Because all it did was confuse her. I mentioned that Wikipedia math pages are a "dick measuring contest for experts on the subject" and the light went on behind her eyes and she laughed and agreed. There are far better resources and I suggested she ask her teacher for them.
Re:Linking to Wikipedia to explain math (Score:5, Interesting)
Because if you look in all other commercial encyclopedias (encyclopediae?), you get a more english (well, natural language) translation of the concepts of a math article. But not even that, Wikipedia on this subject fails even at the post-secondary textbook level. I don't count myself among the dumbest of the population, but when I go to a Wikipedia page for something that is on my level for math, the articles on things like cycloids and such are much better explained by Machinery's Handbook or any other source, really, than there.
I am not saying that Wikipedia should dumb its articles down to the point where even the most innumerate among us would understand all of them, but the "spam equations on the wall with little explanation" model doesn't work very well unless you are immersed in the subject. For example, concepts covered in Algebra I and II in high school should be written for that level.^1 Also, this "write for the grad-student and mathemetician for everything" model does little to help people who use applied mathematics. Indeed, this whole focus on grad-student and up writing in the math articles is at odds with the rest of the Wikipedia.
As a result, anyone wishing to *learn* anything about math is better off using anything but Wikipedia.
Your response to me that the articles are written by grad students and mathemeticians (not all mathemeticians are jerks, btw) for grad students and mathemeticians reinforces the fact that it certainly seems like a giant circle jerk.
The problem is that these topics aren't what you'd see in high school algebra. In fact, upper level undergraduate courses would probably just touch on these. So yes, encyclopedias would have more easily understood articles but they almost certainly don't cover theorems like the ABC theorem or topology in any depth. In fact, most articles in encyclopedias will probably give you a very cursory explanation. To make an analogy it'd be like explaining people as living things with 2 legs, 2 arms and which breath air. It's not useful for any in depth topic and when you really want to understand, you'll need to go into details. And in math, those details come in the form of definitions and equations explaining how the definitions interact together.
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>The problem is that these topics aren't what you'd see in high school algebra.
But the fact is that I was able to pull up a *better* explanation of what ABC triplets are and what this conjecture is by linking to a project dealing directly with this problem run by an actual mathemetician. And it was in terms that anyone in algebra I or pre-algebra, if they slowed down and took it step-by-step, could comprehend and it was accurate.
And if you clicked through to the other pages, on the site, you found clear
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And nothing anyone said defending the Wikipedia math pages contradicts my initial claim that you shouldn't link those pages to explain math, especially to a bunch of non-mathemeticians on Slashdot.
Nobody's really disputing that. You kept reiterating that Wikipedia Math articles are written by jerks. They're most unhappy about that. To support your point, you basically compared the Wikipedia article to a webpage that's simple enough to suit curious junior high pupils, but it is not up to your own opinion whether one shall cater Wikipedia to this level of pedagogy. It is wrong to say something's absolutely bad just because something else is better (for a different purpose). For example, I can easily f
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Until then I'm going to be grateful for the already large amount of time and effort put in to providing what is already there.
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And nothing anyone said defending the Wikipedia math pages contradicts my initial claim that you shouldn't link those pages to explain math, especially to a bunch of non-mathemeticians on Slashdot
Actually, it does. The wikipedia article in this case is a really good example of encyclopedic content. It gives a brief description of the ABC conjecture which is quite easy to understand. It then goes into a little more depth and gives a nice bunch of further reading links.
It's a great refersher for anyone who ma
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I am a mathematician, and I agree most of what you say.
One quibble -- the "dick-measuring contest" claim smacks of conspiracy theory -- the more prosaic and correct reason is that it is far easier for a mathematician to write the true mathematics into a Wiki article than it is for him or her to create a translation suitable for more general audiences.
(May I suggest you could improve the impact of your writing by spelling the word "mathematician" correctly?)
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Your response to me that the articles are written by grad students and mathemeticians (not all mathemeticians are jerks, btw) for grad students and mathemeticians reinforces the fact that it certainly seems like a giant circle jerk.
I wrote a couple of the original pages on wikipedia dealing with some comp-sci type topics not usually taught in a 4 year program (back in ~2000). I thought they were fairly clear and understandable, complete with short pieces of pseudo code, and algorithmic explanations. I gave
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It is by no means a dick measuring contest, at least not in the way you are saying. Math, computer science, and physics are areas I am very well versed in and articles written about those subjects on wikipedia are very easy for me to read and comprehend. Howe
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What you should have told that high school student was that wikipedia is secondary reference material, not a learning aid. If you need help learning a subject matter you should be asking for help from a tutor, teacher, instructor, or educational textbook, not wikipedia.
Then Wikipedia math articles should never *ever* be referred to in a general context to introduce an unfamiliar subject to anyone.
it is ostensibly a databank for knowledge (and crazy admins)
This is a damning indictment of Wikipedia's mission.
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Given that most readers on this website are in the software engineering field who haven't studied advanced mathematics (though a lot probably have), I'd say you're right in this context. If this was posted on a math forum the wikipedia article would probably be an appropriate explanation. Similarly I probably wouldn't link to the wikipedia article "P versus NP problem" on a che
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I am not a mathemetician. I know my way aronud a reasonable bit of maths as part of my day job (engineering of sorts), but I never studies maths apart from in my engineering course and what I have had to pick up from books since.
Ocasionally random bits of maths interest me (like involution matrices) so I read about them. I find wikipedia a very useful resource.
Because if you look in all other commercial encyclopedias (encyclopediae?), you get a more english (well, natural language) translation of the conce
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I mentioned that Wikipedia math pages are a "dick measuring contest for experts on the subject"
Please check out the comment above by exploder (should be easy to find - it is rated +5 Insightful). In particular:
the articles are written in a way that makes them most useful to the people who donate their time to produce them
I just want to briefly provide an example as to why this is a good thing. I'm a math/stats guy. For me, the free and easily accessible Wikipedia pages are always my first port of call when looking into a new topic/method.
On the other side of the coin is my best mate. He is a med science guy. He avoids Wikipedia like it has the plague and instead uses a resource that is behind a paywall. Why? Be
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The historical dead paper encyclopedia wouldn't even have entries on these kinds of things. It is true wikipedia's math articles are written at a graduate or higher level, though. Personally, my response was to stop settling for less than the real deal and become a math grad student - one course at a time (taking my third now).
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But...
ABC triples are not a complex subject. They are arithmetic.
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So an entry about poems, should lust list the rules, and be done. Actually, a lot of knowledge is contained in the examples, the reason why the concept was needed, how it evolved, and ways to make the concept clear to one that isn't in the field but is generally smart. If you ARE in the field, you likely have specialized books. I don't go to Wikipedia for economic or finance related issues. I got to books that I trust and are solid.
So...I don't agree. Math in wikipedia is useless to me. I don't go to wikipe
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Wikipedia is a fantastic first reference for working mathematicians or grad students
As a physics graduate student, I can say this is not my experience. Wikipedia articles are terribly written because an encyclopedia is not the place to learn about concepts.
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Sorry if it wasn't obvious, but I meant grad students in mathematics.
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Although exploder's explanation may be accurate, it in no way justifies the massive uselessness that most of the math articles -- including ones about subjects that are fully capable of being explained to the laymen (as the Encyclopaedia Britannica has done for years).
There is nothing wrong with the technical math being included. There is everything wrong with the intuitive explanations not being included when they are feasible.
The major problem is a bunch of pompous mathematicians (and as a mathematician I
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If you're a mathematician who sees math articles in WP which are missing intuitive explanations that could feasibly be added, then by all means be bold and add them! Just, please, if it's an important one like "manifold", read the talk page first to see if there's already a consensus about the technical level. There's a lot of thought and effort put into striking the right balance that may not be apparent from just reading the articles.
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Re:intuitive interpretation (Score:2)
I think I disagree a little. I see it as unfortunate that the people expressing their frustration walked into the trap of one of the logical fallacies (which one?) of using the Four-Star laden words (NSFW denoted as ****) in doing so. However there is a point under that flawed frustrated presentation. Put in a fancier manner, the rest of Wikipedia is indeed at a generalist level, meant for people who just want to know what something is, and then go back to their life. In those cases, the "Encyclopedia is on
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That's what mathematics is—obscure and non-obvious stuff is everywhere. Sorry if you're only interested in arithmetic, but math is a huge, complicated subject. One of my math professors said that he'd never found an error in a mathematics article in Wikipedia, which is true in my experience.
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>That's what mathematics isâ"obscure and non-obvious stuff is everywhere.
But it doesn't have to be that way. There needs to be more Vi Harts as Wikipedia article editors than Benoit Mandelbrots with communication difficulties (not saying that people like Mandelbrot or Penrose are lacking in communication, Mandelbrot was, and Penrose is a brilliant writer, but rather there appear to be many WP math editors who are "high in clock cycles, low in I/O").
I swear, the difference between people getting tur
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I agree with your perspective 100% on this. I have been reading a lot of WP math articles lately, and many of them are really difficult to follow. A Google search usually brings up a different site within 2-3 results which has a 10x better, easier to read explanation of the same concept. I'm not stupid, I have a 135+ IQ...but blaming the reader and making excuses is par for the course at WP.
The funny thing is, most of these concepts I've read about are at their heart quite simple. It's the people who have n
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The thing is though, there are mathemeticians who are good at writing articles on math that *can* explain things so that people not necessarily immersed in higher math 24/7 *can* understand what is being talked about.
The page I cited earlier was written by a mathemetician running a project called "abcathome" - a BOINC project looking at abc triples. And it's understandable. And it gives you an method how to discover whether 3 numbers are an abc triple. It also doesn't resort to any fancy formulas or jarg
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There are enough books on mathematics that are worth reading by the general public. There's no reason why wikipedia's math section should follow suit. It fills a gap that is visible to, and only affects, highly trained professionals in scientific fields. It's a niche reasource, and as such it's pretty good as it goes.
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Battles?
Well, you know what they say: The penis mightier than the sword.
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The Penis Measuring Armageddon will be fought on Pen Island.
http://www.penisland.net/ [penisland.net]
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BMO
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Yeah, that "new, conceptual universes" line lit up my bullshit detector like a Christmas tree. But the author is well-established, so it's probably a bad translation and/or breathless hype inserted by the university PR office.
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I did not read the article yet, but my guess is that it is some journalist's lame attempt to "explain" category theory to laymen.
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Actually after reading a bit more, it turns out not to be as hyperbolic as it sounds. The author has come up with a whole constellation of new mathematical constructions to support his claimed proof. As the article points out, this means it'll take quite some time for mathematicians to understand these constructions before they'll be able to judge the correctness of the proof. This kind of thing would be dismissed out of hand if it came from Joe Nobody, but Shinichi Mochizuki's reputation in this case sh
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Perhaps the Rhythm of the Primes has a new conductor.
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This kind of thing would be dismissed out of hand if it came from Joe Nobody, but Shinichi Mochizuki's reputation in this case should ensure that it gets a good look.
I wonder how many interesting insights we miss due to such bigotry.
Re:linky whacky (Score:4, Insightful)
Good question. Why don't you devote twenty years or so to becoming competent to judge, then spend all your time reading every crackpot's theory on trisecting angles or why pi isn't really transcendental, and let us know what you find out?
in research mathematics? (Score:3)
Probably extremely few.
A friend of mine knew Shin (as he was known then) when he was an undergraduate. The guy was obscene insane-clown-level genius prodigy. Not the prodigy in the sense of the people who can shoot the lights out of the Putnam Competition but even far deeper than that, and jumping into very difficult and profound concepts by age 17 or 18. He did a small stint doing independent research with Ed Witten before moving up to pure mathematics. By 2nd or 3rd year undergrad (age 17 or so), he wa
This has already been worked on... (Score:5, Funny)
...and solved. I think it was the early (19)70's. A researcher named Jackson
(with the help of his brothers) came to the conclusion that it was simple as 1-2-3.
Additional verification shown that do-re-mi fit the bill as well. At the time, people
were sing all about it - I'm surprised this has come up again.
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Jackson's prior graduate studies were never too much for him to jam into his schedule. Furthermore, his fellow grads were kind enough to leave him alone, so he could learn enough to heal the world and still have time to rock Robin (Billie Jean was not his lover) and make her want to scream.
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A researcher named Jackson (with the help of his brothers) came to the conclusion that it was simple as 1-2-3.
This cannot be truth because spreadsheet software wasn't available until the 80's.
See Peter Woit / Not Even Wrong (Score:5, Informative)
Peter had a pretty good first glance reaction to the paper: http://www.math.columbia.edu/~woit/wordpress/?p=5104 [columbia.edu]
I haven't seen any good discussions of the actual math content of the paper yet though.
Good discussions on math content (Score:4, Informative)
See http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture for a discussion on the mathematical content by experts.
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Sounds like he's working in a bit of a vacuum. That's always high-risk. At least it's out now, so critique can begin. I won't be convinced until Tao, Mazur, Elkies, etc. are convinced.
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This is one of the things I've always hated about the reporting on math, which is not only the fault of reporters but also of mathematicians.
...
But mathematics really needs to get less abstract in its terminology. The name needs to mean something, just like how in CS you call something "method_does_this()" instead of "method_x()".
Well, the names often are meaningful, but after a while one starts running out of words, and/or the concepts just get so specialized that there aren't any words that convey anything close to the right idea of what's happening.
So what do (Score:2)
NBC , CBS, and FOX say about this conjecture?
OOP FTW (Score:1)
thinking of numbers not as members of sets (the standard interpretation), but instead as objects which exist
Of course he was able to solve the problem; he used an Object Oriented framework!
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When you see a cube, you define it's boundaries in this universe by it's sides and edges,
In this theory primes, q 1 make up the "dimensions" if you will.
Want an easier explanation? The abc conjecture is a universal equation through which (seemingly) all other equations can be refactored to make them comparable and translatable. Great for number theorists and programmers, not sure who else will use it. Maybe physicists.
Diophantine textbook... (Score:2)
I find these titbits about number theory absolutely fascinating... I followed a few courses at undergraduate level that touched on this material - without giving me a solid grounding. What I'd like to know is this: Is there a good textbook that would bring me up to speed with this material? I like Wikipedia articles - but I find them disjointed.. what I'd like from a textbook is something that leads me through the subject from undergraduate level onwards. Can anyone make any recommendations?
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I find these titbits about number theory absolutely fascinating... I followed a few courses at undergraduate level that touched on this material - without giving me a solid grounding. What I'd like to know is this: Is there a good textbook that would bring me up to speed with this material? I like Wikipedia articles - but I find them disjointed.. what I'd like from a textbook is something that leads me through the subject from undergraduate level onwards. Can anyone make any recommendations?
I've had pretty good success with Wolfram MathWorld [wolfram.com].
enjoyed a bit of recreational math (Score:2)
Nice article to spur a bit of recreational math. They even have a nice little "quality" formula to use for rating your finds. It's obvious that the place to look is powers of small numbers, especially primes.
I used a few command line tools, bc and factor, and some bash shell scripting to check a few combinations. Skimmed through the results of commands like this:
for ((i=1;i < 25;i++));do echo -n "$i "; echo "13^15-5^$i"|bc|factor;done
With that, I found a few decent quality combinations:
5 + 2^1