I THINK MY HEAD JUST EXPLODED

A Chinese mathematician just proved a new property of prime numbers.

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.
 

zhang

 
Prime numbers — those that have no factors other than 1 and themselves — are the atoms of arithmetic and have fascinated mathematicians since the time of Euclid, who proved more than 2,000 years ago that there are infinitely many of them.

Because prime numbers are fundamentally connected with multiplication, understanding their additive properties can be tricky. Some of the oldest unsolved problems in mathematics concern basic questions about primes and addition, such as the twin primes conjecture, which proposes that there are infinitely many pairs of primes that differ by only 2, and the Goldbach conjecture, which proposes that every even number is the sum of two primes. (By an astonishing coincidence, a weaker version of this latter question was settled in a paper posted online by Harald Helfgott of École Normale Supérieure in Paris while Zhang was delivering his Harvard lecture.)

Prime numbers are abundant at the beginning of the number line, but they grow much sparser among large numbers. Of the first 10 numbers, for example, 40 percent are prime — 2, 3, 5 and 7 — but among 10-digit numbers, only about 4 percent are prime. For over a century, mathematicians have understood how the primes taper off on average: Among large numbers, the expected gap between prime numbers is approximately 2.3 times the number of digits; so, for example, among 100-digit numbers, the expected gap between primes is about 230.

But that’s just on average. Primes are often much closer together than the average predicts, or much further apart. In particular, “twin” primes often crop up — pairs such as 3 and 5, or 11 and 13, that differ by only 2. And while such pairs get rarer among larger numbers, twin primes never seem to disappear completely (the largest pair discovered so far is 3,756,801,695,685 x 2666,669 – 1 and 3,756,801,695,685 x 2666,669 + 1).

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 Y?ld?r?m of Bog(aziçi 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.

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)

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.

 

sieve_of_eratosthenes
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)

 

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.

Much (headachingly much more) at the link.

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3 Responses to I THINK MY HEAD JUST EXPLODED

  1. opaquelypellucid says:

    “…the intrinsic appeal of these conjectures has given them the status of a mathematical holy grail, even though they have no known applications.” That’s all I needed to know. At my age I have enough useless information floating around in my brain and don’t need to add to it. 🙂

  2. bobf says:

    Looks Greek to me.

  3. bogsidebunny says:

    Actually bobf,

    The modern numbering system is of Arabic origin.

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