Results 21  30
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108
Yıldırım, Small gaps between primes or almost primes
"... Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of ex ..."
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Cited by 8 (2 self)
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Abstract. Let pn denote the nth prime. Goldston, Pintz, and Yıldırım recently proved that (pn+1 − pn) lim inf =0. n→ ∞ log pn We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let qn denote the nth number that is a product of exactly two distinct primes. We prove that lim inf n→ ∞ (qn+1 − qn) ≤ 26. If an appropriate generalization of the ElliottHalberstam Conjecture is true, then the above bound can be improved to 6. 1.
Small gaps between primes
"... ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between ..."
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Cited by 7 (3 self)
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ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.
RSA Key Generation with Verifiable Randomness
 In Public Key Cryptography 2002, LNCS 2274
, 2002
"... Abstract. We consider the problem of proving that a user has selected and correctly employed a truly random seed in the generation of her RSA key pair. This task is related to the problem of key validation, the process whereby a user proves to another party that her key pair has been generated secur ..."
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Cited by 7 (1 self)
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Abstract. We consider the problem of proving that a user has selected and correctly employed a truly random seed in the generation of her RSA key pair. This task is related to the problem of key validation, the process whereby a user proves to another party that her key pair has been generated securely. The aim of key validation is to pursuade the verifying party that the user has not intentionally weakened or reused her key or unintentionally made use of bad software. Previous approaches to this problem have been ad hoc, aiming to prove that a private key is secure against specific types of attacks, e.g., that an RSA modulus is resistant to ellipticcurvebased factoring attacks. This approach results in a rather unsatisfying laundry list of security tests for keys. We propose a new approach that we refer to as key generation with verifiable randomness (KEGVER). Our aim is to show in zero knowledge that a private key has been generated at random according to a prescribed process, and is therefore likely to benefit from the full strength of the underlying cryptosystem. Our proposal may be viewed as a kind of distributed key generation protocol involving the user and verifying party. Because the resulting private key is held solely by the user, however, we are able to propose a protocol much more practical than conventional distributed key generation. We focus here on a KEGVER protocol for RSA key generation. Key words: certificate authority, key generation, nonrepudiation, publickey infrastructure, verifiable randomness, zero knowledge 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.
The number of Goldbach representations of an integer
 Proc. Amer. Math. Soc
"... Abstract. Let Λ be the von Mangoldt function and R(n) = ∑ h+k=nΛ(h)Λ(k) be the counting function for the Goldbach numbers. Let N ≥ 2 and assume that the Riemann Hypothesis holds. We prove that N ∑ N 2 ∑ N R(n) = − 2 ..."
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Cited by 6 (2 self)
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Abstract. Let Λ be the von Mangoldt function and R(n) = ∑ h+k=nΛ(h)Λ(k) be the counting function for the Goldbach numbers. Let N ≥ 2 and assume that the Riemann Hypothesis holds. We prove that N ∑ N 2 ∑ N R(n) = − 2
On the Size of the First Factor of the Class Number of a Cyclotomic Field
, 1990
"... We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two wellknown and widely believed conjectures of analytic number theory. 1. Introduction In 1850 Kummer [13] published a review of the ..."
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We show that Kummer's conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field is untrue under the assumption of two wellknown and widely believed conjectures of analytic number theory. 1. Introduction In 1850 Kummer [13] published a review of the main results that he and others had discovered about cyclotomic fields. In this elegant report he claimed that he had found an explicit "law for the asymptotic growth" of h 1 (p), the socalled first factor of the class number of the cyclotomic field, and would provide a proof elsewhere. This proof never appeared and we believe that Kummer's claim is incorrect. More precisely, let p denote any odd prime, let h(p) be the class number of the cyclotomic field Q(i p ) (where i p is a primitive pth root of unity) and h 2 (p) be the class number of the real subfield Q(i p +i \Gamma1 p ). Kummer proved that the ratio h 1 (p) = h(p)=h 2 (p) is an integer which he called the first factor of the ...
On the incompatibility of two conjectures concerning prime numbers
 Proc. Symp. Pure Math. (Analytic Number Theory
, 1972
"... Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a measure of good luck. Several instances of this will emerge in due course, but one example now may give the ide ..."
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Cited by 6 (0 self)
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Introduction. This talk is about the interplay between computers and theoretical research, as experienced by someone who is not a computer expert. The story involves, among other things, a measure of good luck. Several instances of this will emerge in due course, but one example now may give the idea: The speaker and his coworker, Douglas Hensley,
Primes in Tuples I
"... We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. E ..."
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We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the ElliottHalberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, pn+1 − pn lim inf =0. n→ ∞ log pn We will quantify this result further in a later paper (see (1.9) below).
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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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.
Finding prime pairs with particular gaps
 Math. Comp
, 2002
"... Abstract. By a prime gap of size g, we mean that there are primes p and p + g such that the g − 1 numbers between p and p + g are all composite. It is widely believed that infinitely many prime gaps of size g exist for all even integers g. However, it had not previously been known whether a prime ga ..."
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Abstract. By a prime gap of size g, we mean that there are primes p and p + g such that the g − 1 numbers between p and p + g are all composite. It is widely believed that infinitely many prime gaps of size g exist for all even integers g. However, it had not previously been known whether a prime gap of size 1000 existed. The objective of this article was to be the first to find a prime gap of size 1000, by using a systematic method that would also apply to finding prime gaps of any size. By this method, we find prime gaps for all even integers from 746 to 1000, and some beyond. What we find are not necessarily the first occurrences of these gaps, but, being examples, they give an upper bound on the first such occurrences. The prime gaps of size 1000 listed in this article were first announced on the Number Theory Listing to the World Wide Web on Tuesday, April 8, 1997. Since then, others, including Sol Weintraub and A.O.L. Atkin, have found prime gaps of size 1000 with smaller integers, using more ad hoc methods. At the end of the article, related computations to find prime triples of the form 6m +1, 12m − 1, 12m + 1 and their application