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THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY
, 2007
"... We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao ..."
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We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes.
a, isodivision a �ˆb � Iˆ b
"... Dedicated to the 30th anniversary of China reform and opening We establish the Santilli’s isomathematics based on the generalization of the modern mathematics. Isomultiplication a�ˆa � abTˆ ..."
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Dedicated to the 30th anniversary of China reform and opening We establish the Santilli’s isomathematics based on the generalization of the modern mathematics. Isomultiplication a�ˆa � abTˆ
Additive Combinatorics with a view towards Computer Science and Cryptography  An Exposition
, 2011
"... Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is ..."
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Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define – perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of additive structures in sets equipped with a group structure – we may have other structure that interacts with this group structure. This newly emerging field has seen tremendous advances over the last few years, and has recently become a focus of attention among both mathematicians and computer scientists. This fascinating area has been enriched by its formidable links to combinatorics, number theory, harmonic analysis, ergodic theory, and some other branches; all deeply crossfertilize each other, holding great promise for all of them! There is a considerable number of incredible problems, results, and novel applications in this thriving area. In this exposition, we attempt to provide an illuminating overview of some conspicuous breakthroughs in this captivating field, together with a number of seminal applications to sundry parts of mathematics and some other disciplines, with emphasis on computer science and cryptography.
On the Foundamental Theorem in Arithmetic Progession of Primes
"... Using Jiang function we prove the foundamental theorem in arithmetic progression of primes [13]. The primes contain only k < Pg + 1 long arithmetic progressions, but the primes have no k> Pg + 1 long arithmetic progressions.Terence Tao is recipient of 2006 Fields medal.Green and Tao proved th ..."
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Using Jiang function we prove the foundamental theorem in arithmetic progression of primes [13]. The primes contain only k < Pg + 1 long arithmetic progressions, but the primes have no k> Pg + 1 long arithmetic progressions.Terence Tao is recipient of 2006 Fields medal.Green and Tao proved that the primes contain arbitrarily long arithmetic progressions which is absolutely false[49].They do not understand the arithmetic progression of primes [415].
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS
, 2008
"... Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence ..."
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Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the longstanding conjecture that primes contain arbitrarily long arithmetic progressions. There are many accounts and expositions of the GreenTao theorem, including the articles by Kra [119] and Tao [182] in the Bulletin. The purpose of the present article is to survey a broader, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the GreenTao theorem is a famous descendant. An old geometric problem lies at the heart of key conjectures in harmonic analysis. A major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles. An unexpected argument points harmonic analysts towards additive number theory, with consequences that could have hardly been anticipated. We will not
1 The Simplest Proofs of Both Arbitrarily Long Arithmetic Progressions of primes
"... Using Jiang functions 2 ()J ω, 3 ()J ω and 4 ()J ω we prove both arbitrarily long arithmetic progressions of primes: (1) 1 1, n iP P di+ = + 1 ( , ) 1P d =, 1, 2, , 1, 1i k n = − ≥L, which have the same Jiang function; (2) 1 1, 1, 2,, n i gP P i iω+ = + = L 1,k − 1n ≥ ， 2 gg P P Pω ≤ ≤ = Π and gen ..."
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Using Jiang functions 2 ()J ω, 3 ()J ω and 4 ()J ω we prove both arbitrarily long arithmetic progressions of primes: (1) 1 1, n iP P di+ = + 1 ( , ) 1P d =, 1, 2, , 1, 1i k n = − ≥L, which have the same Jiang function; (2) 1 1, 1, 2,, n i gP P i iω+ = + = L 1,k − 1n ≥ ， 2 gg P P Pω ≤ ≤ = Π and generalized arithmetic progressions of primes i gP P iω = + and,nk i gP P iω+ = + 1, , , 2i k n = ≥L. The GreenTao theorem is false, because they do not prove the twin primes theorem and arithmetic progressions of primes [3].