165-year-old math problem on verge of solution

In the twenty-first century, mathematics need not necessarily be a lone enterprise as the recent breakthroughs demonstrate

April 09, 2014 11:36 pm | Updated May 21, 2016 10:00 am IST

Dr. Terence Tao (right), Professor of Mathematics, University of California, Los Angeles started the Polymath 8 project. — PHOTO: M. SRINATH

Dr. Terence Tao (right), Professor of Mathematics, University of California, Los Angeles started the Polymath 8 project. — PHOTO: M. SRINATH

There is news for math enthusiasts: the twin prime conjecture, a 165-year-old problem, is on the verge of being solved. Starting from Chinese mathematician Yitang Zhang’s breakthrough in April 2013, there has been steady progress on the problem.

The twin prime conjecture is that there are infinitely many pairs of prime numbers that are separated by two units — (3, 5), (5, 7), (11, 13), (17, 19) etc.

While Zhang’s work proved that there are indeed infinitely many pairs of prime numbers, the separation between them could not be shown to be two units; it could be as large as 70 million. The most important contribution has been Zhang’s ability to set an upper bound on the separation.

Taking it further, the Polymath 8 project has now reduced the upper bound on the separation from 70 million to 252. It can be further shrunk to just six if another, difficult but plausible, conjecture is assumed.

Scott Morrison first blogged about this problem, and subsequently the Polymath 8 project was set up which crowdsourced the problem.

The power of numbers

The Polymath 8 project was a public collaboration which, as it progressed, drew substantial mathematical contributions as well as computational and programming work, from about two dozen people, according to Terence Tao, Fields medallist and the person who started the Polymath 8 project.

The path was set by the fact that Zhang’s proof has three independent components, in other words, it has three modules. Improving any of these modules would reduce the upper bound of 70 million. Members of the Polymath collaboration set to work following three approaches.

The first approach is to look for certain special narrow patterns of integers which one believes to be likely to capture many primes. Following this path brought the bound down to about 50 million.

The second approach

The second approach is using sieve theory, which can be used if you know about how prime numbers lie in arithmetic progression. The last approach is to compute new ways in which primes “live” in arithmetic progressions.

“It was improvements in the last two aspects that led to the most significant gains, eventually bringing us down to 4,680,” Terence Tao, who is a professor of mathematics at the University of California, Los Angeles, says in an email to this Correspondent.

Meanwhile, working independently, James Maynard, a post-doctoral fellow at the University of Montreal, made another breakthrough.

He had found a way to improve the sieve theory module to shrink the upper bound of Zhang from 70 million to a mere 600.

What was remarkable is that Maynard’s method did away with the third, and most difficult step of Zhang’s proof mentioned above. Independently, Tao, too, arrived at the same result.

In the next attempt, Polymath8b along with James Maynard reduced the upper bound further to 252.

Where we stand

When asked whether this could actually bring down the bound all the way to two, thereby proving the twin prime conjecture, Tao is doubtful. “Unfortunately there is not much further scope for improvement with our current methods.

Although we can get all the way down to six if we assume an additional, unproven conjecture known as the Elliott-Halberstam conjecture,” he noted in his email.

Mathematician Ram Murty of Queen’s University, Canada, says in an email, “This recent work is significant because we had not fully understood the power of the methods discovered half a century ago…

It is quite possible that within a few years we will get to N=2; perhaps one more new idea is needed, but already, considerable insight has been gained by grappling with the problem in this way.”

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