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To use the Sieve of Eratosthenes to find, say, all the primes up to , 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. 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. 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.

Any amount more would be enough. But the more researchers tried to overcome this obstacle, the thicker the hair seemed to become. Meanwhile, Zhang was working in solitude to try to bridge the gap between the GPY result and the bounded prime gaps conjecture. During the difficult years in which he was unable to get an academic job, he continued to follow developments in the field.

After three years, however, he had made no progress. To take a break, Zhang visited a friend in Colorado last summer. While the new sieve allowed Zhang to prove that there are infinitely many prime pairs closer together than 70 million, it is unlikely that his methods can be pushed as far as the twin primes conjecture, Goldston said. Even with the strongest possible assumptions about the value of the level of distribution, he said, the best result likely to emerge from the GPY method would be that there are infinitely many prime pairs that differ by 16 or less. It took Zhang several months to work through all the details, but the resulting paper is a model of clear exposition, Granville said.

Once Zhang received the referee report, events unfolded with dizzying speed. Invitations to speak on his work poured in. For Zhang, who calls himself shy, the glare of the spotlight has been somewhat uncomfortable. Honsberger, R. Mathematical Gems II. Washington, DC: Math. Kraitchik, M. New York: W. Norton, pp. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. Lehmer, D. List of Prime Numbers from 1 to Washington, DC: Carnegie Institution, Moser, L.

On the Sum of Consecutive Primes. Nagell, T. New York: Wiley, pp. Number Theory and Its History. New York: Dover, Pappas, T. Peano, G. Torino, Italy, Ramachandra, K. Indian Nat. Part A 64 , , Ribenboim, P. The Little Book of Big Primes.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Salamin, E.

Primes and Zeros: A Million Dollar Mystery. Hilbert predicted it would be solved within a few years. A century later, the Clay Mathematics Institute identified. making statements at international conferences, just solved one of the most Prime numbers are the building-blocks of our numbers, because every The primes are thus the basic elements of our entire number system we.

Item 53 in Beeler, M. Schroeppel, R. Item 29 in Beeler, M. Schlafly, R. December 13, Sloane, N. The Encyclopedia of Integer Sequences. Tietze, H. New York: Graylock Press, pp. Torelli, G. Naples, Italy: Tip. Tropfke, J. Geschichte der Elementar-Mathematik, Band 1. Berlin: p. Wagon, S. Freeman, pp.

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Weisstein, E. Wells, D. Middlesex, England: Penguin Books, Zaiger, D. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

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Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. MathWorld Book. Terms of Use. Comparing the Primes in Base 2.

Prime Number A prime number or prime integer, often simply called a "prime" for short is a positive integer that has no positive integer divisors other than 1 and itself. The first few primes are illustrated above as a sequence of binary bits.

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