Major progress in prime number theory

The Green-Tao theorem resolves an important special case of the Erdös-Turan conjecture

Kumbakonam: Professor Terence Tao of the University of California, Los Angeles (UCLA), was awarded the 2006 SASTRA's Ramanujan Prize at the International Conference on Number Theory and Combinatorics at the Srinivasa Ramanujan Centre, SASTRA University, Kumbakonam.

This $10,000 prize comes on the heels of the Fields Medal that was awarded to Professor Tao in August for revolutionary contributions to several areas of mathematics.

Following the award ceremony on Ramanujan's birthday at Kumbakonam, Professor Tao delivered the Ramanujan Commemoration Lecture entitled "Long arithmetic progressions of primes," in which he reported major progress in prime number theory based on his recent work with Professor Ben Green of Cambridge University.

One of the most famous unsolved problems in mathematics is the Prime Twins Conjecture, which asserts that there are infinitely many prime pairs that differ by 2. More generally, the prime k-tuples conjecture states that if a k-tuple is admissible, then there are infinitely many such k-tuples of primes. Here by admissible one means that the k-tuple must satisfy certain non-divisibility conditions.

If the prime k-tuples conjecture is true, then it follows that there are arbitrarily long arithmetic progressions of primes. For example, 7, 37, 67, 97, 127, 157, is an arithmetic progression of 6 primes with common difference 30.

Sieve theory was developed in the 20th century to attack problems such as the k-tuples conjecture. Although this conjecture is still unsolved, sieve methods have succeeded in establishing similar results for almost primes, namely, those integers with very few prime factors, but not for the primes themselves.

Thus, the world was astonished when Professor Tao and Professor Green proved in 2003 that there are arbitrarily long arithmetic progressions of primes. The road to the Green-Tao theorem has been long, and in his lecture, Professor Tao surveyed the history of the problem and described the techniques that led to the recent breakthrough.

The first major advance was made in 1939 by van der Corput, who showed that there are infinitely many triples of primes in arithmetic progression. He used the circle method, originally invented by Hardy and Ramanujan to estimate the number of partitions of an integer and subsequently improved by Hardy and Littlewood to apply to a wide class of problems in additive number theory.

van der Corput's result was improved in 1981 by the British mathematician Heath Brown, who showed that there are infinitely many quadruples in arithmetic progression of which three are primes, and the fourth an almost prime with at most two prime factors. That such an improvement came after more than 40 years indicates the difficulty of the problem.

Another problem was the study of finite arithmetic progressions within sets of positive density. This was pioneered by the 1958 Fields medallist K.F. Roth, who in 1956 showed that any set of integers with positive density contains infinitely many triples in arithmetic progression. This study culminated in 1975 with the grand result of the Hungarian mathematician Szemeredi, who proved that any set of integers with positive density contains arithmetic progressions of arbitrary length. Professor Tim Gowers of Cambridge University, who won the Fields Medal in 1994, has recently given a simpler proof of Szemeredi's theorem. It is to be noted that since the primes have zero density, Szemeredi's theorem does not imply that there are arbitrarily long arithmetic progressions of primes.

Professor Green was a Ph.D student of Professor Gowers, who introduced him to Szemeredi's theorem. One of Professor Green's first major accomplishments was the result that any subset of the primes, which has relative positive density, contains infinitely many triples on arithmetic progressions. Professor Tao and Professor Green then corresponded due to their common interest on such problems. They studied the general problem of arithmetic progressions in sparse sets of integers. By combining ideas from ergodic theory, the techniques of Professor Gowers, and repeated use of Szemeredi's theorem, they were able to prove the astonishing result that there are arbitrarily long arithmetic progressions of primes. The ingredients of the proof were put together when Professor Green visited Professor Tao at UCLA in 2003.

The great Hungarian mathematicians Paul Erdös and Paul Turan conjectured that if A is an infinite set of integers the sum of whose reciprocals is divergent, then there are arbitrarily long arithmetic progressions in A. Since the sum of the reciprocals of the primes is a divergent series, the Green-Tao theorem is a special case of the Erdös-Turan conjecture, which remains unsolved in full generality. Erdös has offered $10,000 for a resolution of this conjecture. The Green-Tao theorem resolves an important special case of the Erdös-Turan conjecture and is a phenomenal achievement by two brilliant young mathematicians. Thus, it was a fitting tribute to Ramanujan that this great work was presented in his hometown on his birthday.

A solution to one of the most difficult problems in mathematics was the most important advance of 2006

A solution to one of the most difficult problems in mathematics was the most important advance of 2006, according to the prestigious journal Science.

Grigory Perelman's proof of the century-old Poincare Conjecture has caused a sensation, and not just because of the brilliance of the work.

In August, the Russian became the first person to turn down a Fields Medal, the highest honour in mathematics.

He also seems likely to turn down a $1m prize offered by a US maths institute.

Dr Perelman is said to despise self-promotion and describes himself as isolated from the rest of the mathematical community.

The best piece of mathematics we have seen in the last 10 years Terence Tao, UCLA

But his work has set the field alight with excitement - and controversy.

Terence Tao, professor of mathematics at the University of California, Los Angeles, called Perelman's result "the best piece of mathematics we have seen in the last 10 years".

Timofey Shilkin, a former colleague of Perelman at the Steklov Mathematics Institute in St Petersburg, Russia, told BBC News: "He definitely deserves the Fields Medal - that is my personal opinion. I am completely sure he is a genius."

'Excellent mathematician'

He added: "I'm afraid he is quite a self-enclosed person. We know about him approximately the same as you know - not too much.

"I met him when he was a member of our group and our contacts were about once a week, but we had only short discussions.

Grigory Perelman shuns the spotlight

"I know nothing about his personal life; I know only that he is an excellent mathematician."

The reclusive Dr Perelman left the Steklov Institute in January, and was last said to be unemployed and living with his mother in her apartment in St Petersburg.

For several years he worked, for the most part, alone on the Poincare Conjecture. Then, in 2002, he posted on the internet the first of three papers outlining a proof of the problem.

The Poincare is a central question in topology, the study of the geometrical properties of objects that do not change when they are stretched, distorted or shrunk.

The surface of the Earth is what topology describes as a two-dimensional sphere. If one were to encircle it with a lasso of string, it could be pulled tight to a point.

On the surface of a doughnut, however, a lasso passing through the hole in the centre cannot be shrunk to a point without cutting through the surface.

Checking the work

Since the 19th Century, mathematicians have known that the sphere is the only enclosed two-dimensional space with this property; but they were uncertain about objects with more dimensions.

The Poincare Conjecture says that a three-dimensional sphere is the only enclosed three-dimensional space with no holes.

Proof of the Conjecture eluded mathematicians until Perelman posted his work on the website arXiv.org.

This is a so-called pre-print server, where researchers upload study papers for informal feedback before they submit them to a peer-reviewed journal.

Feuding within the mathematical community now threatens to overshadow Dr Perelman's achievement.

The Russian had detailed a way to kick down the roadblock that had stymied a solution to the problem. It was then up to others to check his proof.

It was at this stage of the process - when mathematicians pored over Perelman's work to assess its accuracy - that much bad feeling started to rise to the surface.

'Complete proof'

In 2005, a Chinese team consisting of Huai-Dong Cao of Lehigh University and Xi-Ping Zhu of Zhongshan University published what they claimed was "the first written account of a complete proof of the Poincare Conjecture".

Cao and Zhu took on the task of checking Perelman's proof at the behest of their mentor Shing-Tung Yau, a Chinese-born professor of mathematics at Harvard University, US.

2006 saw progress in understanding Neanderthal DNA (Copyright: Natural History Museum)

Shortly after the Cao-Zhu paper was published, Professor Yau gave a speech in which he was reported as having said: "In Perelman's work, many key ideas of proofs are sketched or outlined, but complete details of the proofs are often missing."

This drew the ire of others in the field, who said that Yau's promotion of his proteges' work went too far.

In a rare interview, Perelman told the New Yorker magazine: "It is not clear to me what new contribution did they make."

However, speaking to the New York Times newspaper in October, Professor Yau denied having said there were gaps in Dr Perelman's work.

Science magazine also named its "breakdown" of the year: the scandal involving South Korean cloning pioneer Hwang Woo-suk, whose report of the production of stem cells from a cloned human embryo was found to have been faked.