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The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
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A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
Long arithmetic progressions of primes
 Mathematics Proceedings
"... Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes ..."
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Abstract. This is an article for a general mathematical audience on the author’s work, joint with Terence Tao, establishing that there are arbitrarily long arithmetic progressions of primes. 1. introduction and history This is a description of recent work of the author and Terence Tao [11] on primes in arithmetic progression. It is based on seminars given for a general mathematical
Arithmetic progressions and the primes  El Escorial lectures
 Collectanea Mathematica (2006), Vol. Extra., 3788 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial
"... Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1. ..."
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Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.
Seven consecutive primes in arithmetic progression
 Math.Comp
, 1997
"... Abstract. It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. In 1967, the first such sequence of 6 consecutive primes in arithmetic progression was found. Searching for 7 consecutive primes in arithmetic progression is difficult because it ..."
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Abstract. It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. In 1967, the first such sequence of 6 consecutive primes in arithmetic progression was found. Searching for 7 consecutive primes in arithmetic progression is difficult because it is necessary that a prescribed set of at least 1254 numbers between the first and last prime all be composite. This article describes the search theory and methods, and lists the only known example of 7 consecutive primes in arithmetic progression. 1.
A new algorithm for constructing large Carmichael
 Ken Nakamula, Department of Mathematics and Information Sciences, Tokyo Metropolitan University, MinamiOsawa, Hachioji
, 1996
"... Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 11015 ..."
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Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 1101518 factors. 1.
AN AMAZING PRIME HEURISTIC
"... The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: ..."
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The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes:
ANEWALGORITHM FOR CONSTRUCTING LARGE CARMICHAEL NUMBERS
"... Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk −1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 110151 ..."
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Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk −1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 1101518 factors. 1.