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Progress in Prime Progressions
3 May 2004 (All day)
Two mathematicians have made a stunning breakthrough in the theory of prime numbers--at least that's the preliminary assessment of experts who are looking at their complicated 50-page proof.
Two years ago, Ben Green and Terence Tao began working on the problem of arithmetic progressions of primes: sequences of primes (numbers divisible only by themselves and 1) that differ by a constant amount. One such sequence is 13, 43, 73, and 103. In 1939, a Dutch mathematician proved that there are infinitely many arithmetic progressions of primes with three terms, such as 3, 5, 7 or 31, 37, 43. Green and Tao hoped to prove the same result for four-term progressions. The theorem they got, though, proved the result for prime progressions of all lengths.
Green, who is currently at the Pacific Institute of Mathematical Sciences in Vancouver, British Columbia, and Tao, at the University of California, Los Angeles, started with a 1975 theorem by Endre Szemerédi of the Hungarian Academy of Sciences. Szemerédi proved that arithmetic progressions of all lengths crop up in any "positive fraction" of the integers--basically, any subset of integers whose ratio to the whole set doesn't dwindle away to zero as the numbers get larger and larger. Primes violate that condition. So Green and Tao set out to show that Szemerédi's theorem still holds when the integers are replaced with a smaller set of numbers with special properties and then to prove that the primes constitute a positive fraction of that set.
To build their set, they applied a branch of mathematics known as ergodic theory (loosely speaking, a theory of mixing or averaging) to mathematical objects called pseudorandom numbers. Pseudorandom numbers are not truly random, because they are generated by rules, but they behave like random numbers for certain mathematical purposes. Using these tools, Green and Tao constructed a pseudorandom set of primes and "almost primes," numbers with relatively few prime factors compared to their size.
The last step, establishing the primes as a positive fraction of their pseudorandom set, proved elusive. Then Andrew Granville, a number theorist at the University of Montreal, pointed Green to some work on the size of gaps between primes, some of which proved tailor-made for Green and Tao's research.
The paper, which has been submitted to the Annals of Mathematics, is many months from acceptance. "The problem with a quick assessment of it is that it straddles two areas," Granville says. Even so, says Green's former adviser Timothy Gowers at Cambridge University, who received the 1998 Fields Medal, mathematics' equivalent of the Nobel Prize, for work on related problems, "it's a very, very spectacular achievement."