A Prime Sieve
Instead, it showed that there will always be pairs of primes much closer together than the average spacing predicts. More precisely, GPY showed that for any fraction you choose, no matter how tiny, there will always be a pair of primes closer together than that fraction of the average gap, if you go out far enough along the number line. But the researchers couldn’t prove that the gaps between these prime pairs are always less than some particular finite number.
GPY uses a method called “sieving” to filter out pairs of primes that are closer together than average. Sieves have long been used in the study of prime numbers, starting with the 2,000-year-old Sieve of Eratosthenes, a technique for finding prime numbers.
To use the Sieve of Eratosthenes to find, say, all the primes up to 100, start with the number two, and cross out any higher number on the list that is divisible by two. Next move on to three, and cross out all the numbers divisible by three. Four is already crossed out, so you move on to five, and cross out all the numbers divisible by five, and so on. The numbers that survive this crossing-out process are the primes.
The Sieve of Eratosthenes works perfectly to identify primes, but it is too cumbersome and inefficient to be used to answer theoretical questions. Over the past century, number theorists have developed a collection of methods that provide useful approximate answers to such questions.
“The Sieve of Eratosthenes does too good a job,” Goldston said. “Modern sieve methods give up on trying to sieve perfectly.”
GPY developed a sieve that filters out lists of numbers that are plausible candidates for having prime pairs in them. To get from there to actual prime pairs, the researchers combined their sieving tool with a function whose effectiveness is based on a parameter called the level of distribution that measures how quickly the prime numbers start to display certain regularities.
The level of distribution is known to be at least ½. This is exactly the right value to prove the GPY result, but it falls just short of proving that there are always pairs of primes with a bounded gap. The sieve in GPY could establish that result, the researchers showed, but only if the level of distribution of the primes could be shown to be more than ½. Any amount more would be enough.
The seeds of Zhang’s result lie in
a paper from eight years ago that
number theorists refer to as GPY, after its three authors — Goldston, János
Pintz of the Alfréd Rényi Institute of Mathematics in Budapest, and Cem Yıldırım
of Boğaziçi University in Istanbul. That paper came tantalizingly close but was
ultimately unable to prove that there are infinitely many pairs of primes with
some finite gap.
Instead, it showed that there will always be pairs of primes much closer together than the average spacing predicts. More precisely, GPY showed that for any fraction you choose, no matter how tiny, there will always be a pair of primes closer together than that fraction of the average gap, if you go out far enough along the number line. But the researchers couldn’t prove that the gaps between these prime pairs are always less than some particular finite number.
GPY uses a method called “sieving” to filter out pairs of primes that are closer together than average. Sieves have long been used in the study of prime numbers, starting with the 2,000-year-old Sieve of Eratosthenes, a technique for finding prime numbers.
To use the Sieve of Eratosthenes to find, say, all the primes up to 100, start with the number two, and cross out any higher number on the list that is divisible by two. Next move on to three, and cross out all the numbers divisible by three. Four is already crossed out, so you move on to five, and cross out all the numbers divisible by five, and so on. The numbers that survive this crossing-out process are the primes.
The Sieve of Eratosthenes This
procedure, which dates back to the ancient Greeks, identifies all the primes
less than a given number, in this case 121. It starts with the first prime —
two, colored bright red — and eliminates all numbers divisible by two (colored
dull red). Then it moves on to three (bright green) and eliminates all multiples
of three (dull green). Four has already been eliminated, so next comes five
(bright blue); the sieve eliminates all multiples of five (dull blue). It moves
on to the next uncolored number, seven, and eliminates its multiples (dull
yellow). The sieve would go on to 11 — the square root of 121 — but it can stop
here, because all the non-primes bigger than 11 have already been filtered out.
All the remaining numbers (colored purple) are primes. (Illustration: Sebastian
Koppehel)
The Sieve of Eratosthenes works perfectly to identify primes, but it is too cumbersome and inefficient to be used to answer theoretical questions. Over the past century, number theorists have developed a collection of methods that provide useful approximate answers to such questions.
“The Sieve of Eratosthenes does too good a job,” Goldston said. “Modern sieve methods give up on trying to sieve perfectly.”
GPY developed a sieve that filters out lists of numbers that are plausible candidates for having prime pairs in them. To get from there to actual prime pairs, the researchers combined their sieving tool with a function whose effectiveness is based on a parameter called the level of distribution that measures how quickly the prime numbers start to display certain regularities.
The level of distribution is known to be at least ½. This is exactly the right value to prove the GPY result, but it falls just short of proving that there are always pairs of primes with a bounded gap. The sieve in GPY could establish that result, the researchers showed, but only if the level of distribution of the primes could be shown to be more than ½. Any amount more would be enough.
