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.
1 comment:
What's "prime number theory"? What's "number theory" or "prime number",for that matter,and who really cares anyways? An opinion rife among common people is that all such stuffs are meant for nerds engrossed in their own impenetrable universe,blissfully occupied in abstract pursuits devoid of the slightest practical value!
An expositor is most wanting here:Allow me to dispel those outrageous misconceptions:-In the first place,a terse and grossly abridged definition of Number theory (or The Theory of Numbers,formerly called "The Higher Arithmetic")is :a branch of Pure Mathematics concerned with investigating the properties of whole numbers(i.e.,unearthing general underlying laws which families of numbers or numbers in general comply with). The great Gauss spoke of Number Theory in these terms:'Mathematics is the queen of Science,and Number Theory is the queen of Mathematics'; Another quotation from Leopold Kronecker goes thus:'God created the integers,the rest is the work of men.'
True enough,being a fundamental science,basic number theory is concerned with sheer knowledge acquisition and primarily serves no other purpose or constraints with regards to practical applications. Ironically,however,since the advent of computers in the late 60's,as we find our lives steeped into the digital age,quite stunningly the purest among Pure Mathematics is virtually changing its status towards that of an applied science:Indeed,no secure e-payment transaction today can make do with those Number Theoretic priciples underlying Public Key Crytography(the science of coding and decoding information);likewise, no nationwide defence ministry can afford to ignore the latest developments in algorithmic and computational number theory so that tapped or intercepted top secret messages may not get easily deciphered.
So,what do we mean when we say a number is prime? In elementary Chemistry courses,subtances are classified into the elements and compounds dichotomy,the latter themselves being composed of elements,albeit,chemically bound and in differing proportions. Likewise,a similar behaviour is evinced among the natural numbers:If we attempt to split numbers into product of smaller numbers,two grand families segregate;those splittable or decomposable into product of smaller numbers,and those which can't be,except trivially.The former are called composite numbers(e.g.,520=2x2x2x5x13; 4719=3x11x11x13);the latter are the prime numbers or simply primes(e.g.,523=1x523 ;4721=1x4721). Prime numbers being exclusively the multiple of themselves,apart from 1,they cannot hence be factored any further. Instead,they serve as building blocks or elements for producing larger composite numbers. Hence the paramount importance in identifying them;they are,actually,the gems scattered among the natural numbers.
Now,since the 115 or so elements of the Mendeleev periodic table build up any imaginable substance,however complex,an intuitive guess would be that only a group of finite primes also build all the natural numbers,however large. Observations subtantiate that the primes indeed thin out among the large numbers. As far back as the time of Euclid(Greek mathematician who lived in the Vth century AD),however,this question has been settled by an ingenious 'reductio ad absurdum' argument(i.e.,proof by contradiction),which definitely establishes that,in fact,there can be no greatest prime,i.e.,even if they decline in number as we move higher along the number line,Primes Never Entirely Cease. At this point,the analogy with Chemistry collapses.
Two apparently opposite properties characterise the primes; What is most intriguing and mind-boggling with the lore of primes is that although no hard and fast rule exists that predicts the occurence of a prime(there is no largest possible distance between two consecutive primes),yet it has been firmly established (discovered by Carl Friedrich Gauss and proved jointly by Jacques Hadamard and Charles de la Vallee Poussin) that statistically,the N-th prime is N times the natural logarithm of N(or lnN),or equivalently,in the neighbourhood of large enough N,the average distance between consecutive primes is lnN. This asymptotic result regarding the distribution of primes (i.e.,the frequency of large primes varies inversely as their logarithm) is referred to as the Prime Number Theorem or PNT. It has been shown also to bear a close link with and forms a natural consequence of the celebrated Riemann's hypothesis:A most unexpected connexion has been observed,over the past decades, between prime distribution and the spectral energy states exhibited at quantum scales in atoms and responsible for quantum chaos. Some authors have stretched their beliefs to the extent that they are convinced the very mind of God lurks within the distribution of primes !
What really is the Twin Prime Conjecture? Apart from 2,all primes are odd(1 is not considered a prime),and hence form a subset of odd numbers. Whilst odd numbers occur regularly with a fixed difference of 2,primes spring eratically among them. It may,nonetheless,happen that two successive odd numbers be both prime(e.g.,17,19 or 101,103),in which case we say they form a twin prime pair. Although the infinitude of primes is an established fact,the infinitude or otherwise of twin prime pairs remains a pending issue;it is surmised that this set is infinite but no rigorous proof has ever been provided;this tough problem has been called the Twin Prime Conjecture. A major breakthrough which helped make a quantum leap towards sketching a possible future proof is a result in 2003 on small gaps between primes by D.Goldston & C.Yildirim,confirmed in 2005 after patching a flaw in the original work.(http://aimath.org/primegaps)
As an extension of Szemeredi's theorem,the Green-Tao theorem of 2004 is another bold stunning result concerning the sequence of primes which states that there exists arbitrarily long progressions of primes.
Number theory is the one single branch of mathematics that is most close to fun and recreational topics. Yet,quite often seemingly unassuming questions turn out to be intractable. And especially so in additive number theory with prime connected issues where conjectures abound such as the Goldbach conjecture,the Legendre conjecture,the Mersenne prime conjecture,the Gilbreath conjecture,the de Polignac conjecture,the Giuga conjecture,etc.
The great Hungarian mathematician Paul Erdos once summarised the situation in these words:"Mathematics is not ready to tackle such questions. It might as well take a few centuries of refinement" ! Oh boy! Primes are such a lure !
Post a Comment