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Updated: May 24, 2013 03:10 IST

China’s Ramanujan finds prime number gap

Shubashree Desikan
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Yitang Zhang achieves breakthrough in solving the longstanding problem of twin prime conjecture

There has been a breakthrough in number theory, with Chinese mathematician Yitang Zhang coming up with a result that is seen as a step towards solving the twin prime conjecture, a longstanding problem in mathematics.

And, coming up from relative obscurity and humble origins as he does, the 50-year-old shares elements of the story of Srinivasa Ramanujan closer home in India.

What this lecturer of the University of New Hampshire in the United States has solved is very close to the famous twin primes problem: the question whether there are infinitely many pairs of consecutive prime numbers that differ by 2.

The paper has been accepted for publication by Annals of Mathematics in record time — three weeks since it was submitted. It will be published in a forthcoming issue.

Dr. Zhang has come up with proof that there is an infinite number of consecutive prime numbers that are separated by a gap bounded by 70 million. Thus, he has shown that there is a number N smaller than 70 million such that there are infinitely many prime numbers that differ by N. It is a long way off from 2 to 70 million, but the important thing is that the gap is not infinite but a known number — 70 million. This is a major step towards solving, or as a mathematician would put it, “weakening,” the twin prime conjecture.

Says D. Surya Ramana, mathematician of the Harish-Chandra Research Institute, Allahabad: “Zhang’s result is marvellous because it is for the first time that we know that there are infinitely many pairs of prime numbers (p, q) with p > q and p-q bounded by a fixed constant, which in the case of Zhang is around 70 million, which is of course very large, but that does not all detract from the importance of the result.”

Dr. Zhang has shot to fame in Ramanujan-like fashion. A Chinese immigrant in the U.S., he received his doctorate from Purdue University. His dissertation was not even on prime numbers. After his Ph.D, he was unable to get an academic job but continued to think about numbers as he worked as an accountant and even in a ‘Subway’ sandwich outlet, until he got the job at the University of New Hampshire. His dedication finally paid off; after three years of labouring over the problem, the answer came to him when he took a break, shortly before leaving for a concert.

Srinivasa Ramanujan (1887-1920), from Tamil Nadu, had almost no formal training in pure mathematics and was an office worker in Madras (now Chennai), but made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions.

Says R. Balasubramaniam, number theorist and Director of the Institute of Mathematical Sciences, Chennai: “[Professor Zhang’s] is a fantastic result. A major step in this direction was taken by Goldston, Pintz and Yildrim eight years ago, so this result would have been expected. But not this soon — this is stupendous!”

Professor Yitang Zhang must be congratulated on this remarkable achievement. As Professor Balasubramanian comments, "this was expected but not this soon!" Indeed, the experts know of the milestone accomplishments in analytic number theory of the last 50 years that are integral in Professor Zhang's solution. Indeed, if we are to see further, we must "stand on the shoulders of giants". There are two essential ingredients needed to make such an advance. First is technical skill and the second is faith in one's own abilities to push ahead and ignore the prevailing dogmas of the pundits. Professor Zhang has demonstrated both of these qualities in this breakthrough. Even more impressive is that Zhang's PhD is in commutative algebra, far removed from classical analytic number theory. Clearly, there are elements of "Ramanujan" in this story! Well done!

from:  Ram Murty
Posted on: May 26, 2013 at 23:51 IST

Can some one explain any practical implication of this finding? Is there any real use with this
finding?
I am very eager to understand..

from:  Raj
Posted on: May 24, 2013 at 21:55 IST

In my opinion, Zhang have solve this with a new approach to this
problem. Although this will help in many other areas but making the gap
N(=70 million)->Infinity will take different approach altogether.
(Applying Mathematical Induction to approaches). This proof is a closed
chapter, nonetheless its an very important feat as it paved the way for
future work.

from:  Santosh
Posted on: May 24, 2013 at 20:53 IST

Anand, my namesake, you are gravely mistaken. There are infinite prime pairs with
infinite gap between them (N -> infinite) is not the aim here. The aim here is to
prove that N = 2 (or closest to it). We are getting closer and closer.

from:  Anand
Posted on: May 24, 2013 at 18:38 IST

Comparison with Srinivasa Ramanujan is not proper. Mr Zhang is closer to modern math academic-- a specialist focusing on one problem or a set of problems. (I used to be one of them.) Ramanujan was a self-taught creative genius who could think up literally hundreds of problems that others had not dreamed existed. These have kept mathematicians busy nearly a hundred years after Ramanujan's death.

This is to take nothing away from Dr. Zhang, a formally trained mathematician from a top university (Purdue) who by all accounts has taken a significant step in solving an important problem in number theory. While commendable, this does not make him a prodigy and prolific genius like Ramanujan but an outstanding mathematical problem solver.

from:  N.S. Rajaram
Posted on: May 24, 2013 at 17:45 IST

The author is struggling to tie this story to Ramanujam. It would be better if the article jsut reported about the current issue.

from:  SRA
Posted on: May 24, 2013 at 17:18 IST

Great! Being a number theorist, I worked on the same problem by giving a
weaker proof. However,Professor Zhang’s given an exceptional idea to
close the problem. I wish him good luck.

from:  Prof. Dr. K. Raja Rama Gandhi
Posted on: May 24, 2013 at 15:16 IST

Anand, you are missing the point. The challenge is to reduce N to 2 not
infinity.

from:  Giridhar
Posted on: May 24, 2013 at 12:25 IST

The above does not take care of CONSECUTIVE primes!
It is also necessary to specify:

modulo[(p+q),2]=0.

N =70 million has been proved.

The next challenge is to extend N to a larger number.

The ultimate would be N-> infinity.
When the number system is unbounded, why does it seem difficult to do
that? Let mathematicians worry, I am not one.

from:  Anand
Posted on: May 24, 2013 at 10:41 IST

While admiring the findings of Dr.Zhang, we need not rush to make comparison with Ramanujan.That has been our tendency and by this mental adjustment, we have a satisfaction that we are not lesser than anybody else. If we escalate the comparison, there are countless other mathematicians.

from:  P.Radhakrishnan
Posted on: May 24, 2013 at 09:53 IST

Great, especially problems like these which can be explained to High School students, but do not have a proof are difficult to solve. However, to quote that he worked in a subway or was an accountant, leading a very modest life etc. I think that is more of popularity gaining tactics or could be media hype. This man who is in his 50's has a PhD in Mathematics and has been working as a lecturer for quite some time at University of New Hampshire, teaching undergrad classes in the Mathematics department. In US universities, teaching alone is not a very hectic job. You have lot of leisure time. And, when asked what other problems he is working on, he does not want to tell! Ramanujan neither had a PhD nor the luxury of a permanent job and he was barely in his 30's. The int'l community has done a good job to recognize this lecturer of Mathematics, but to compare him with the life and works of a prodigy like Ramanujan is an injustice to the great Indian mathematician of all times.

from:  Srini
Posted on: May 24, 2013 at 05:40 IST
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