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Beauty of Prime Numbers

It’s a result only a mathematician could love. Researchers hoping to get ‘2’ as the answer for a long-sought proof involving pairs of prime numbers are celebrating the fact that a mathematician has wrestled the value down from infinity to 70 million.

Last week, Yitang Zhang, a popular math professor at the University of New Hampshire, stunned the world of pure mathematics when he announced that he had proven the “bounded gaps” conjecture about the distribution of prime numbers—a crucial milestone on the way to the even more elusive twin primes conjecture, and a major achievement in itself.

A number is prime if you can't divide it by anything but 1 and itself. A twin prime is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (821, 823), etc. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin. The largest known twin primes are 3,756,801,695,685 × 2^666,669 + 1 and 3,756,801,695,685 × 2^666,669 - 1, and were discovered in 2011.

Every positive number can be expressed in just one way as a product of prime numbers. For instance, 60 is made up of two 2s, one 3, and one 5. The primes are the atoms of number theory, the basic indivisible entities of which all numbers are made. As such, they’ve been the object of intense study ever since number theory started. One of the very first theorems in number theory is that of Euclid, which tells us that the primes are infinite in number; we will never run out, no matter how far along the number line we let our minds range.

But mathematicians are greedy types, not inclined to be satisfied with mere assertion of infinitude. After all, there’s infinite and then there’s INFINITE. There are infinitely many powers of 2, but they’re very rare. Among the first 1,000 numbers, there are only 10 powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512.
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There are infinitely many even numbers, too, but they’re much more common: exactly 500 out of the first 1,000. In fact, it’s pretty apparent that out of the first X numbers, just about (1/2)X will be even.

Primes, it turns out, are intermediate—more common than the powers of 2 but rarer than even numbers. Among the first X numbers, about X/log(X) are prime; this is the Prime Number Theorem, proven at the end of the 19th century by Hadamard and de la Vallée Poussin. This means, in particular, that prime numbers get less and less common as the numbers get bigger, though the decrease is very slow; a random number with 20 digits is half as likely to be prime as a random number with 10 digits.

Naturally, one imagines that the more common a certain type of number, the smaller the gaps between instances of that type of number. If you’re looking at an even number, you never have to travel farther than 2 numbers forward to encounter the next even; in fact, the gaps between the even numbers are always exactly of size 2. For the powers of 2, it’s a different story. The gaps between successive powers of 2 grow exponentially, and there are finitely many gaps of any given size; once you get past 16, for instance, you will never again see two powers of 2 separated by a gap of size 15 or less.

Those two problems are easy, but the question of gaps between consecutive primes is harder. It’s so hard that, even after Zhang’s breakthrough, it remains a mystery in many respects.

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states There are infinitely many primes p such that p + 2 is also prime. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics. In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs p and q such that q - p = 2k. The case k = 1 is the twin prime conjecture.

"In number theory in particular, conjectures are pretty understandable," says Henryk Iwaniec of Rutgers University in Piscataway, New Jersey. "But proving is another matter."

To make their work a little easier, mathematicians have aimed at answering a slightly different question: is there an infinite number of primes which have a neighbouring prime some finite distance away, even if that distance is much larger than 2?

"My main result is just this: yes," said Yitang Zhang of the University of New Hampshire in Durham at a seminar at Harvard University yesterday.

“That’s only [a factor of] 35 million away” from the target, quips Dan Goldston, an analytic number theorist at San Jose State University in California . “Every step down is a step towards the ultimate answer.”

A major milestone was reached in 2005 when Goldston and two colleagues showed that there is an infinite number of prime pairs that differ by no more than 16 . But there was a catch. “They were assuming a conjecture that no one knows how to prove,” says Dorian Goldfeld, a number theorist at Columbia University in New York.

The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever. The jump from 2 to 70 million is nothing compared with the jump from 70 million to infinity. “If this is right, I’m absolutely astounded,” says Goldfeld.

"These values are very rough," Zhang says. "I think to reduce them to less than one million or even smaller is very possible" – although mathematicians may need another breakthrough to reduce the distance all the way down to just 2 and finally prove the twin prime conjecture.

Iwaniec is less concerned about that problem at the moment, though. "The 70 million is not very important," he says. What matters is that Zhang was able to show that the gap between adjacent primes cannot exceed a certain value. "People will be stunned by the result. I'm sure people will be working on it for years and then bring it down eventually."

Iwaniec, who has made contributions to the twin prime problem but was not involved in the new work, has reviewed a paper presenting Zhang's proof and cannot find an error in it. Zhang's paper has been accepted for publication in the Annals of Mathematics.

Goldston, who was sent a copy of the paper, says that he and the other researchers who have seen it “are feeling pretty good” about it. “Nothing is obviously wrong,” he says. Goldston does not think the value can be reduced all the way to 2 to prove the twin prime conjecture. But he says the very fact that there is a number at all is a huge breakthrough. “I was doubtful I would ever live to see this result,” he says.

Despite the apparent simplicity of the bounded gaps conjecture, Zhang’s proof requires some of the deepest theorems of modern mathematics, like Pierre Deligne’s results relating averages of number-theoretic functions with the geometry of high-dimensional spaces.

"His result is beautiful," Iwaniec says. "He should enjoy his 15 minutes of fame."

[by Dhiraj Sarmah.

Courtesy : 1. "First proof that infinitely many prime numbers come in pairs" URL : http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 ;
2. "Proof that an infinite number of primes are paired " URL : http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html?full=true&print=true
3. "The Beauty of Bounded Gaps" URL : http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_
primes_conjecture_a_huge_discovery_about_prime_numbers.html
4. Wikipedia]


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