When arranging the natural numbers in a spiral and emphasizing the prime numbers, an intriguing and not fully explained pattern is observed, called the Ulam spiral.
Image: Flickr/Center for Image in Science and Art _ UL
From Nature magazine
The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved.
Mathematician Shinichi Mochizuki of Kyoto University in Japan has released a 500-page proof of the abc conjecture, which proposes a relationship between whole numbers — a 'Diophantine' problem.
The abc conjecture, proposed independently by David Masser and Joseph Oesterle in 1985, might not be as familiar to the wider world as Fermat’s Last Theorem, but in some ways it is more significant. “The abc conjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat's Last Theorem,” says Dorian Goldfeld, a mathematician at Columbia University in New York. “If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the twenty-first century.”
Like Fermat’s theorem, the abc conjecture refers to equations of the form a+b=c. It involves the concept of a square-free number: one that cannot be divided by the square of any number. Fifteen and 17 are square free-numbers, but 16 and 18 — being divisible by 42 and 32, respectively — are not.
The 'square-free' part of a number n, sqp(n), is the largest square-free number that can be formed by multiplying the factors of n that are prime numbers. For instance, sqp(18)=2×3=6.
If you’ve got that, then you should get the abc conjecture. It concerns a property of the product of the three integers axbxc, or abc — or more specifically, of the square-free part of this product, which involves their distinct prime factors. It states that for integers a+b=c, the ratio of sqp(abc)r/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a=3 and b=125, so that c=128, then sqp(abc)=30 and sqp(abc)2/c = 900/128. In this case, in which r=2, sqp(abc)r/c is nearly always greater than 1, and always greater than zero.
Deep connection
It turns out that this conjecture encapsulates many other Diophantine problems, including Fermat’s Last Theorem (which states that an+bn=cn has no integer solutions if n>2). Like many Diophantine problems, it is all about the relationships between prime numbers. According to Brian Conrad of Stanford University in California, “it encodes a deep connection between the prime factors of a, b and a+b”.
Many mathematicians have expended a great deal of effort trying to prove the conjecture. In 2007, French mathematician Lucien Szpiro, whose work in 1978 led to the abc conjecture in the first place claimed to have a proof of it, but it was soon found to be flawed.
Like Szpiro, and also like British mathematician Andrew Wiles, who proved Fermat’s Last Theorem in 1994, Mochizuki has attacked the problem using the theory of elliptic curves — the smooth curves generated by algebraic relationships of the sort y2=x3+ax+b.
There, however, the relationship of Mochizuki’s work to previous efforts stops. He has developed techniques that very few other mathematicians fully understand and that invoke new mathematical ‘objects’ — abstract entities analogous to more familiar examples such as geometric objects, sets, permutations, topologies and matrices. “At this point, he is probably the only one that knows it all,” says Goldfeld.
Conrad says that the work “uses a huge number of insights that are going to take a long time to be digested by the community”. The proof is spread across four long papers1–4, each of which rests on earlier long papers. “It can require a huge investment of time to understand a long and sophisticated proof, so the willingness by others to do this rests not only on the importance of the announcement but also on the track record of the authors,” Conrad explains.
Mochizuki’s track record certainly makes the effort worthwhile. “He has proved extremely deep theorems in the past, and is very thorough in his writing, so that provides a lot of confidence,” says Conrad. And he adds that the pay-off would be more than a matter of simply verifying the claim. “The exciting aspect is not just that the conjecture may have now been solved, but that the techniques and insights he must have had to introduce should be very powerful tools for solving future problems in number theory.”
This article is reproduced with permission from the magazine Nature. The article was first published on September 10, 2012.
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32 Comments
Add CommentTypo with superscript loss, 42 should be 4^2 or 4*4
Reply | Report Abuse | Link to thisText should be:
... Fifteen and 17 are square free-numbers, but 16 and 18 — being divisible by 4^4 and 3^3, respectively — are not.
...
an+bn=cn
Reply | Report Abuse | Link to thisshould be
a^n + b^b = c^n
as presented the article is unreadable.
<blockquote>Fifteen and 17 are square free-numbers, but 16 and 18 — being divisible by 42 and 32, respectively — are not.</blockquote>
Reply | Report Abuse | Link to thisWhat an error! 16 / 42 = 0.38095238095238095238095238095238...
And 18 / 32 = 0.5625
Perhaps we should try dividing by the square roots of the stated numbers?
sqrt(42)= 6.480740698407860230965967436088 NOPE, that's not an integer either!
sqrt(32)= 5.6568542494923801952067548968388 AGAIN not an integer!
This mathematics related Post needs to take a few minutes and watch Kahn Academy or something!
The reason why 16 is not squarefree is because it's divisible by 4, which is equal to 2*2
The reason why 18 is not squarefree is because it's divisible by 9, which is equal to 3*3!
Check your Math!
The explanation is already provided by JamesMoore: the superscripts got lost; '42' should read 4^2 and '32' should read 3^2. 16 is not square-free because it is divisible by 4^2 (16) and 18 is not square-free because it is divisible by 3^2 (9). Read it that way and it makes sense.
Reply | Report Abuse | Link to thisFollow the link at the end of the article to read the text with the superscripts.
Reply | Report Abuse | Link to thisAmong other things, it's interesting that they refer to this as the "abc" Conjecture. Notre that a, b and c are related by a + b = c. So abc is really (a^2)b + a(b^2). There's no need to invoke a third value! Also, the conjecture itself is misstated. First, for a,b > 0, which is assumed, sqp((a^2)b + a(b^2)) will always vbe positive or greater than zero, so sqp((a^2)b + a(b^2))^r/((a^2)b + a(b^2)) must always be greater than zero. In fact, the actual conjecture states that, for any r > 1, sqp((a^2)b + a(b^2))^r/((a^2)b + a(b^2)) > 1, for all but a finite number of values, a and b.
Reply | Report Abuse | Link to thisScientific American, please find a way to publish articles that have fewer errors and can accommodate such things as superscripts and subscripts. It appears that your site is "machine" generated code, which is acceptable as long as any errors in the code are corrected. Sentences such as "Fifteen and 17 are square free-numbers, but 16 and 18 — being divisible by 42 and 32, respectively — are not." really make no sense and make reading the article harder to read as we try to decipher what was really meant to be said.
Reply | Report Abuse | Link to thisIt's even more wrong than simply having lost the superscript formatting. 16 being div by 4^2 is trivial since 4^2 = 16. I think what is meant is 2^2, which also makes more sense when talking about prime factors, 2 being prime, 4 not.
Reply | Report Abuse | Link to thisGive them a break, guys. The fact that there are some errors in the HTML notwithstanding, Mr. Ball has presented a difficult concept in a way that is digestible to non-math majors, like me, and even I immediately understood what Mr. Moore points out.
Reply | Report Abuse | Link to thisAnd beyond that, this is a huge story.
Actually msadesign, as pointed out by Mark R above, Mr Ball has "ballsed it up". Surely we can hope for better than this from SA?
Reply | Report Abuse | Link to thisFor all who are looking for a better reading of this story, try the "Nature" version at: www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378
Reply | Report Abuse | Link to thisThis article is terrible. Even I, who is noooo mathematician, figured out within seconds that 2x3 + 4x3 = 6x3.
Reply | Report Abuse | Link to thisAs someone pointed out above, it should be like in Nature:
Fermat’s Last Theorem (which states that a^n+b^n=c^n has no integer solutions if n>2).
How can it be "digestible" with errors?
Reply | Report Abuse | Link to thisBy going to the actual paper, instead of reading a synopsis of it, written in code to describe another code. HTML has limits and everyone so far has been able to suggest different ways of showing it that they themselves would have understood but not necessarily anyone else. Thus the link to the original, so that if one is intrigued they can read it as it was meant to be read.
Reply | Report Abuse | Link to thiswater: point taken. And still a huge story. It's the latching onto trivia that is so infuriating. Yes, we expect SA to be right. But for pete's sake this story is a lot more important than a few typos, ins't it?
Reply | Report Abuse | Link to thisAnd the image, which isn't explained, really, is fascinating. The idea that numbers, when graphically placed just so, expose themselves in novel ways is…well, I don't know. This entire story in fact makes me think that we are on the verge of something Really Big.
After all isn't it just plain peculiar that the universe is mathematically describable? And that numbers, odd things that they are can have inherent, hidden, useful, and relationships? That they can be inherently predictive?
Don't we conceptualize numbers as mere lifeless tools? What magic awaits when we learn they have hidden character and connections, associations not realized. It's like finding a new dimension in some ways.
It's a great time to be alive.
@msadesign, While I do truly enjoy most of the articles on the web version of Sciam there has been a consistent lack of professionalism in the way they have been presented. It has been common to see html formatting code shown in the printed text rather than what was intended. This was not a browser problem but a problem with the design. Now that I have reread the article I can see that they have indeed made some corrections and are now using superscripts to represent powers. These errors don't make me mad but do make their published content harder to read and we expect better from them. Kudo's to them for taking the mild criticism to heart and correcting the article.
Reply | Report Abuse | Link to thisWell, formatting of article text is probably important but, I think, even more important is article wrong content.
Reply | Report Abuse | Link to thisThis is especially true for such precise science as mathematics.
The "abc" conjecture stated in the article is missing crucial condition that makes it simply incorrect.
And I am not talking about post made by julianpenrod about sqp(abc)^r/c is always greater than 0 because a,b and c are greater than 0 -- here he just misread the statement (sqp(abc)r/c has some minimum value > 0 depended on r only).
I am talking about missing condition of a,b,c being mutually prime numbers. Without that condition, the statement is incorrect.
For example, if a=3 and b=125, so that c=128, then sqp(abc)=30 and sqp(abc)2/c = 900/128 is above 1 but if a=3*2^n and b=125*2^n, so that c=128*2^n, then still sqp(abc)=30 but sqp(abc)^2/c = 900/128*2^n and could be as close to 0 as possible.
Square free numbers are perhaps better explained as being formed by multiplying the prime factors of the number without repeating a factor. So 18 = 2 * 3 * 3 has its sqp(18) = 2 * 3 = 6. But this is a great little story.
Reply | Report Abuse | Link to thisSo wait: does this mean Marilyn vos Savant actually was right about Wiles' proof of FLT? And did the phone company really kill Kennedy?
Reply | Report Abuse | Link to thisObviously, everyone here is rushing to read the original papers and will be reporting back shortly on all the typos. Can't wait.
"And the image, which isn't explained, really, is fascinating. The idea that numbers, when graphically placed just so, expose themselves in novel ways is…well, I don't know."
Reply | Report Abuse | Link to thisSadly, THIS was something my late wife Jean E. Hockin was fascinated by, and spent a lot of time working on.
Almost every article that I read on this site is followed by comments that criticize SA and heap contumely upon it. And there seems to be a lot of vituperative commentary that is barely worthy of the basest rubbish that appears on Twitter. Sadly, these sorts of posts seem to be almost the norm these days. Just chill, guys.
Reply | Report Abuse | Link to thisUnless the article has been re-posted since yesterday, maybe some of you just need a better browser. On my monitor all the superscripts rendered perfectly. (Mozilla Firefox)
Reply | Report Abuse | Link to thisThis article does not explain how the image was generated. You can put almost any formula into the fractal program and generate an image. Some images are visually interesting and some are not. By simply changing the height and width of the canvas almost any group of numbers will form an image. Some will generate the signs of the zodiac just like like the night sky as viewed by ancient astronomers. I see animals in the clouds all the time. Patterns and messages from God must hidden in all the old prime numbers.
Reply | Report Abuse | Link to thisFermat's last theorem ignores the issue of dimensionality: if looking at 2 we have 2 variables a^2+b^2=c^2 will have integer solutions; if we look at 3 variables a^3+b^3+c^3=d^3 will have integer solutions etc...essentially if we have n variables then a^n+b^n+c^n...+[nth term]^n = z^n will have integral solutions.
Reply | Report Abuse | Link to thisShould be pretty obvious that a^n+b^n=c^n will never have integral solutions above its dimensionality (2).
Thank you very much EBGuest - you saved me the trouble of searching the internet for an explanation of what the abc conjecture really says, as the version stated in the article - with or without typos - and even the correction by julianpenrod left me with something that I could immediately disprove by counterexample. And my math knowledge is approximately high school level.
Reply | Report Abuse | Link to thisWith the added criterion, I sure can't think of any counterexample right off my head - and I assume nobody else has come up with any yet, or the article would have a different headline. :)
....Too complicated for me
Reply | Report Abuse | Link to this2 and 4 are all its factors, there is no requirement of prime
Reply | Report Abuse | Link to this///and also like British mathematician Andrew Wiles, who proved Fermat’s Last Theorem in 1994, Mochizuki has attacked the problem using the theory of elliptic curve
Reply | Report Abuse | Link to this////////////////
what does this sentense "Mochizuki has attacked the problem" mean?
is "the problem" here meaning "abc conjection" or attack the way Wiles proved Fermats theorem?
agree, the article has not yet explain the picture exactly.
Reply | Report Abuse | Link to thisUnfortunately, this publication is sloppy and gets sloppier all the time. I see no indication of any will to do anything about it.
Reply | Report Abuse | Link to thisWhen you consider what is "digestible" by the normal standards, me thinks that msdesigns is use to digesting that which is forced as acceptable
Reply | Report Abuse | Link to thisI don't see what the big deal is. I've already come up with a more generalized version of this proof that extends the pairs concept to all integers, not just the primes.
Reply | Report Abuse | Link to thisIt's a very elegant proof but I don't have room to fit it in the margin of this comment.