By: Natalie Wolchover
Published: 09/11/2012 09:17 AM EDT on Lifes Little Mysteries
A Japanese mathematician claims to have the proof for the ABC conjecture, a statement about the relationship between prime numbers that has been called the most important unsolved problem in number theory.
If Shinichi Mochizuki's 500-page proof stands up to scrutiny, mathematicians say it will represent one of the most astounding achievements of mathematics of the twenty-first century. The proof will also have ramifications all over mathematics, and even in the real-world field of data encryption.
The ABC conjecture, proposed independently by the mathematicians David Masser and Joseph Oesterle in 1985 but not proven by them, involves the concept of square-free numbers, or numbers that cannot be divided by the square of any number. (A square number is the product of some integer with itself). According to the mathematics writer Ivars Peterson in an article for the Mathematical Association of America, the square-free part of a number n, denoted by sqp(n), is the largest square-free number that can be obtained by multiplying the distinct prime factors of n. Prime numbers are numbers that can only be evenly divided by 1 and themselves, such as 5 and 17.
The ABC conjecture makes a statement about pairs of numbers that have no prime factors in common, Peterson explained. If A and B are two such numbers and C is their sum, the ABC conjecture holds that the square-free part of the product A x B x C, denoted by sqp(ABC), divided by C is always greater than 0. Meanwhile, sqp(ABC) raised to any power greater than 1 and divided by C is always greater than 1. [What Makes Pi So Special?]
This conjecture may seem esoteric, but for mathematicians, it's deep and ubiquitous. "The ABC conjecture is amazingly simple compared to the deep questions in number theory," Andrew Granville, a mathematician at the University of Montreal, was quoted as saying in the MAA article. (Granville worked at the University of Georgia at the time.) "This strange conjecture turns out to be equivalent to all the main problems. It's at the center of everything that's been going on."
The conjecture has also been described as a sort of grand unified theory of whole numbers, in that the proofs of many other important theorems follow immediately from it. For example, Fermat's famous Last Theorem (which states that an+bn=cn has no integer solutions if n>2) follows as a direct consequence of the ABC conjecture.
In a 1996 article in The Sciences, the mathematician Dorian Goldfeld of Columbia University said the ABC conjecture "is more than utilitarian; to mathematicians it is also a thing of beauty. Seeing so many Diophantine problems unexpectedly encapsulated into a single equation drives home the feeling that all the subdisciplines of mathematics are aspects of a single underlying unity.
"No wonder mathematicians are striving so hard to prove it – like rock climbers at the base of a sheer cliff, exploring line after line of minute cracks in the rock face in the hope that one of them will offer just enough purchase for the climbers to pick their way to the top."
And now, one such climber may have reached the summit. According to Nature News, Mochizuki, a mathematician at Kyoto University, has proved extremely deep theorems in the past, lending credence to his claim that he has the proof for ABC. However, a huge investment of time by many other mathematicians will be required to go through the gargantuan proof and verify the claim.
"If the ABC conjecture yields, mathematicians will find themselves staring into a cornucopia of solutions to long-standing problems," Goldfeld wrote.
- 5 Seriously Mind-Boggling Math Facts
- Mathematicians Want to Say Goodbye to Pi
- Why Southern Cicadas Emerge In Exact Prime Number Cycles
Correction: This article has been updated to reflect that Andrew Granville was at the University of Georgia when he made the quoted statement, as he now is at the University of Montreal.
Also on HuffPost:
The Monkey and the Hunter The <a href="http://www.google.com/url?sa=t&rct=j&q=monkey and hunter&source=web&cd=6&ved=0CFsQFjAF&url=http%3A%2F%2Fbuphy.bu.edu%2F~duffy%2Fsemester1%2Fc04_monkeyhunter.html&ei=iB75Tr_HNYfTiAL-ltCFDQ&usg=AFQjCNHrgX0aj5yuH9JxlyPi-xREdluKHg&cad=rja" target="_hplink">Boston University department of Physics website</a> puts it thus: <blockquote>"A hunter spies a monkey in a tree, takes aim, and fires. At the moment the bullet leaves the gun the monkey lets go of the tree branch and drops straight down. How should the hunter aim to hit the monkey? 1. Aim directly at the monkey 2. Aim high (over the monkey's head) 3. Aim low (below the monkey)"</blockquote> The result may be counterintuitive; gravity acts on the monkey and the bullet at the same rate, so no matter how fast the bullet is going (controlling for air resistance, among other things) the hunter should start by aiming at the monkey. In case you're not convinced, try <a href="http://www.waowen.screaming.net/revision/force&motion/mandh.htm" target="_hplink">this simulation</a>. Photo: Flickr: BinaryApe
Newton's Cannonball In this thought experiment, we're meant to imagine a cannon (elevated high enough so that its projectile will avoid hitting anything on Earth) that fires its cannonball at a 90 degree angle to the Earth below it. The diagram above shows several possibilities for the cannonball's flight, depending on how fast it's going at the moment of launch. If it's too slow, it will eventually fall back down to Earth. If it's too fast, it will escape Earth's gravitation entirely and head out into space. If it's somewhere in the middle, it will be sent into orbit. This realization was a landmark in the study of gravitation, and laid the groundwork for satellites and space flight.
<a href="http://analysis.oxfordjournals.org/content/43/1/33.full.pdf" target="_hplink">Kavka's Toxin Puzzle</a>: <blockquote>"An eccentric billionaire places before you a vial of toxin that, if you drink it, will make you painfully ill for a day, but will not threaten your life or have any lasting effects. The billionaire will pay you one million dollars tomorrow morning if, at midnight tonight, you intend to drink the toxin tomorrow afternoon. He emphasizes that you need not drink the toxin to receive the money; in fact, the money will already be in your bank account hours before the time for drinking it arrives, if you succeed. All you have to do is. . . intend at midnight tonight to drink the stuff tomorrow afternoon. You are perfectly free to change your mind after receiving the money and not drink the toxin."</blockquote> Is it possible to intend to drink the toxin? We're not sure. There's an interesting discussion on the puzzle <a href="http://jsomers.net/blog/toxin" target="_hplink">here</a>. Photo: Flickr: The University of Iowa Libraries
<a href="http://web.archive.org/web/20060831124229/http://www.newyorker.com/archive/content/articles/060619fr_archive01" target="_hplink">Molyneux's Problem</a> <blockquote>"Suppose a man born blind, and now adult, and taught by his touch to distinguish between a cube and a sphere of the same metal, and nighly of the same bigness, so as to tell, when he felt one and the other, which is the cube, which is the sphere. Suppose then the cube and the sphere placed on a table, and the blind man made to see: query, Whether by his sight, before he touched them, he could now distinguish and tell which is the globe, which the cube? To which the acute and judicious proposer answers: 'Not. For though he has obtained the experience of how a globe, and how a cube, affects his touch; yet he has not yet attained the experience, that what affects his touch so or so, must affect his sight so or so...'"</blockquote> Philosopher John Locke, who referenced the problem in his 'Essay On Human Understanding,' agreed, but the thought experiment lay essentially unsolved until last year, when MIT Professor of Vision and Computational Neuroscience Pawan Sinha led a study of patients whose blindness had been reversed. The results agreed with Molyneux's original hypothesis.
<a href="http://books.google.com/books?id=Yfo3rnt3bkEC&pg=PA21&lpg=PA21&dq="If+we+placed+a+living+organism+in+a+box"&source=bl&ots=-dbzGJt86Y&sig=TBI9HJi4Ux4uCU5TW0EXowoMQVs&hl=en&sa=X&ei=XYH5TuT6E9LoiALru_inDg&ved=0CGAQ6AEwCA#v=onepage&q="If we placed a living organism in a box"&f=false" target="_hplink">Twin Paradox</a> Einstein gave the basic formulation as follows: <blockquote>"If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism, the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light."</blockquote> But what if the two organisms happened to be twins? This helps us realize that either one could think of the other as the "traveler," but if that's the case then why has one aged normally and one quickly? It's not quite a "paradox" in the traditional sense of a logical contradiction, but in Einstein's time it was pretty odd. It's been resolved (the traveling twin experiences two instances of acceleration with regard to the stationary twin--one on the way out and one on the way back--that justify the asymmetrical aging), but it's still interesting to think about, if only to imagine how the twins must feel when they meet. Photo: Getty
Flat-Land In the video above, the great science educator and astrophysicist Carl Sagan gives a thought experiment meant to illustrate the incomprehensibility of higher dimensions to lower-dimensional beings. We'll let him speak for himself.
Feynman Sprinkler If you were to force water through a sprinkler with spouts angled, say, clockwise, the sprinkler head would rotate counterclockwise. But what happens if you built a "reverse sprinkler," or a device with the same construction that sucked water in instead of shooting it out? This was only a thought experiment until Physicist Richard Feynman sought to test it (he didn't come up with it) during undergrad at Princeton, and before his rig exploded he found that there was no motion in the reversed version. Stumped? There's a discussion of a real—albeit air-driven—Feynman Sprinkler <a href="http://web.mit.edu/Edgerton/www/FeynmanSprinkler.html" target="_hplink">here</a>.
Galileo's Ship This thought experiment envisions a subject performing various actions and observing various creatures in a closed room in a ship, and then performing the same actions and making the same observations when the ship is in motion at a constant velocity. The full version, too long to reproduce here, can be found at <a href="http://en.wikipedia.org/wiki/Galileo's_ship#The_proposal" target="_hplink">this link</a>. Galileo's discovery—that it's not velocity but acceleration that changes the trajectory of a thrown ball, say, or the flight of a bird—was ahead of its time. It wouldn't be fully utilized for centuries, when Einstein used it to help formulate his theory of special relativity.
Quantum Immortality and Quantum Suicide The video above, titled 'Quantum Immortality,' is a basic illustration of one of the more disturbing thought experiments. In the original formulation, the unlucky subject pulls the trigger of a gun, rigged with a subatomic mechanism that has a 50% chance of activating the bullet, and dies if the gun fires. This hypothetical process is known as quantum suicide. In the many-worlds interpretation of quantum mechanics, there's a world in which the subject lives and one in which he or she dies. A branching point is created at each pull of the trigger; eventually, no matter how many shots are taken, there will be a version of the subject in some world who has survived every shot. He or she is said to have attained quantum immortality.