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ON SUMS OF PRIMES AND TRIANGULAR NUMBERS
 JOURNAL OF COMBINATORICS AND NUMBER THEORY 1(2009), NO. 1, 65–76.
, 2009
"... We study whether sufficiently large integers can be written in the form cp + Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2 a p−r)/m+Tx with p a prime ..."
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Cited by 10 (9 self)
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We study whether sufficiently large integers can be written in the form cp + Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2 a p−r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with related analysis and numerical data. One of our conjectures states that each natural number n ̸ = 216 can be written in the form p + Tx with x ∈ Z and p a prime or zero; another conjecture asserts that any odd integer n> 3 can be written in the form p + x(x + 1) with p a prime and x a positive integer.
Affine linear sieve, expanders, and sumproduct
"... This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and ..."
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Cited by 10 (2 self)
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This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and
Tameness in Expansions of the Real Field
, 2001
"... What might it mean for a firstorder expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and realanalytic geometers to the ominimal setting: expansions of the real field in which every definable set has finitely many c ..."
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Cited by 9 (1 self)
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What might it mean for a firstorder expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and realanalytic geometers to the ominimal setting: expansions of the real field in which every definable set has finitely many connected components. But there are expansions of the real field that define sets with infinitely many connected components, yet are tame in some welldefined sense (e.g., the topological closure of every definable set has finitely many connected components, or every definable set has countably many connected components). The analysis of such structures often requires a mixture of modeltheoretic, analyticgeometric and descriptive settheoretic techniques. An underlying idea is that firstorder definability, in combination with the field structure, can be used as a tool for determining how complicated are given sets of real numbers. This paper is based on a lecture that I delivered at Logic Colloquium '01 (Vienna). I thank the organizers for inviting me to address the Colloquium. Global conventions. Throughout, m, n and p denote arbitrary elements of N (the nonnegative integers). Given a firstorder structure M, with underlying set M , "definable" (in M) means "definable in M with parameters from M " unless otherwise noted. If no ambient space M n is specified, then "definable set" means "definable subset of some M n ". I use "reduct" and "expansion" in the sense of definability, that is, given structures M 1 and M 2 with underlying set M , I say that M 1 is a reduct of M 2 equivalently, M 2 is an expansion of M 1 , or M 2 expands M 1 if every set definable in M 1 is definable in M 2 . For the most part, we shall be concerned with the definable sets of a struc...
Checking the odd Goldbach conjecture up to 10 20
 Math. Comp
, 1998
"... Abstract. Vinogradov’s theorem states that any sufficiently large odd integer is the sum of three prime numbers. This theorem allows us to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed us to check this conje ..."
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Cited by 7 (1 self)
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Abstract. Vinogradov’s theorem states that any sufficiently large odd integer is the sum of three prime numbers. This theorem allows us to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed us to check this conjecture up to 10 20. 1.
A complete Vinogradov 3primes theorem under the Riemann hypothesis
 ERA Am. Math. Soc
, 1997
"... Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1. ..."
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Cited by 6 (1 self)
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Abstract. We outline a proof that if the Generalized Riemann Hypothesis holds, then every odd number above 5 is a sum of three prime numbers. The proof involves an asymptotic theorem covering all but a finite number of cases, an intermediate lemma, and an extensive computation. 1.
Obstructions to uniformity, and arithmetic patterns in the primes, preprint
"... Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to iden ..."
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Cited by 5 (3 self)
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Abstract. In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing in [26] that the primes contain arbitrarily long arithmetic progressions. One of the driving philosophies is to identify precisely what the obstructions could be that prevent the primes (or any other set) from behaving “randomly”, and then either show that the obstructions do not actually occur, or else convert the obstructions into usable structural information on the primes. 1.
Generalising the HardyLittlewood method for primes
 In: Proceedings of the international congress of mathematicians
, 2007
"... Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the ..."
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Cited by 5 (2 self)
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Abstract. The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the HardyLittlewood method has been generalised to obtain, for example, an asymptotic for the number of 4term arithmetic progressions of primes less than N.
An invitation to additive prime number theory
, 2004
"... The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sam ..."
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Cited by 4 (0 self)
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The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.
TERNARY GOLDBACH PROBLEM FOR THE SUBSETS OF PRIMES WITH POSITIVE RELATIVE DENSITIES
, 2007
"... Abstract. Let P denote the set of all primes. Suppose that P1, P2, P3 are three subsets of P with d P (P1) + d P (P2) + d P (P3)> 2, where d P (Pi) is the lower density of Pi relative to P. We prove that for sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. 1. ..."
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Cited by 3 (3 self)
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Abstract. Let P denote the set of all primes. Suppose that P1, P2, P3 are three subsets of P with d P (P1) + d P (P2) + d P (P3)> 2, where d P (Pi) is the lower density of Pi relative to P. We prove that for sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. 1.