Famous prime number conjecture one step closer to proof
New research shows that there are infinite prime numbers separated by a distance of up to 70 million, which could make it possible to solve the twin prime conjecture.
Tue, May 21, 2013 at 08:30 AM
Infinity down, only 69,999,997 to go.
New research has proven that prime numbers don't just disappear as numbers get larger — instead, there is an infinite number of prime numbers separated by a distance of at most 70 million.
The new proof, accepted this month for publication in the journal Annals of Mathematics, takes the field one step closer to solving the twin prime conjecture, a famous mathematical idea that suggests the existence of an infinite number of prime numbers separated by a distance of 2 (for example, the prime numbers 11 and 13, which are separated by 2). Prime numbers are those that are divisible by only themselves and 1.
Prior to this discovery, mathematicians suspected there were infinitely many twin primes, or prime numbers separated by two, but proofs hadn't set bounds on how far apart primes could be separated. [The 9 Most Massive Numbers in Existence]
"It's a huge step forward in terms of showing that there are primes close together," said Daniel Goldston, a mathematician at San Jose State University in California. "It's a big huge step toward the twin prime conjecture."
Other mathematicians also applauded the achievement, and its author, Yitang Zhang, a mathematician unknown in the field. "Basically, no one knows him," said Andrew Granville, a number theorist at the Université de Montréal, as quoted by the Simons Foundation. "Now, suddenly, he has proved one of the great results in the history of number theory."
Simple observation … tough solution
In the 1800s, mathematician Alphonse de Polignac noticed a strange trend in prime numbers. Though so-called twin primes get less common as numbers get bigger, de Polignac became convinced that there were infinitely many twin primes.
But proving it was another matter.
These problems "are very attractive to people because the problems themselves are not difficult to understand, but the solution — the proof — could be very difficult," said Zhang of the University of New Hampshire.
Many attempts relied on finding primes using sieve methods, which essentially involves crossing out numbers that have larger and larger factors to find primes (for instance, crossing out all the numbers divisible by 2, then 3, then 5, then 7, and so on).
All of the small primes can be manually calculated, and if numbers get large enough, mathematicians can generalize the technique. But in between small numbers and big ones is a vast terrain where primes are too big calculate with the sieve, but too small to make generalizations about.
In 2005, Daniel Goldston, a mathematician at San Jose State University in California, and his colleagues János Pintz and Cem Yildirim developed a new method (called GPY) to make claims for that middle range of numbers in order to prove that the numerical gaps between prime numbers are bounded, and not infinite.
"Our method got right up to the point where you would approach getting this bounded gaps result, but we couldn't get it," Goldston said.
Crossing the gap
Zhang had been trying to find a way to close the gap in the GPY method for years. But last summer, he felt a breakthrough was close and devoted all his efforts to cracking the prime problem.
He finally developed set of new mathematical methods and used them to overcome the gap in prior work.
The math community hasn't thoroughly scrutinized the proof to ensure it's airtight, but several mathematicians in the field have done a first-pass check and found the logic sound.
The current known maximum gap between primes is 70 million, but that number may come down dramatically with further iterations of the proof.
Still, it's unlikely that the same methods could be used to prove the twin prime conjecture, Goldston said.
"We are pretty sure these methods aren't going to get down to two," Goldston said. "You have to have some new ideas."
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