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Some Conjectures on the Gaps between Consecutive Primes
 in the Distribution of Prime Numbers ", in Proc. of the 8th Joint EPSAPS Int.Conf. Physics
"... Five conjectures on the gaps between consecutive primes are formulated. One expresses the number of twins below a given bound directly by ß(N ). These conjectures are compared with the computer results and a good agreement is found. 1. Introduction. In 1922 Hardy and Littlewood [1] have proposed ab ..."
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Five conjectures on the gaps between consecutive primes are formulated. One expresses the number of twins below a given bound directly by ß(N ). These conjectures are compared with the computer results and a good agreement is found. 1. Introduction. In 1922 Hardy and Littlewood [1] have proposed about 15 conjectures. The conjecture B of their paper states: There are infinitely many primes pairs (p; p 0 ), where p 0 = p + d, for every even d. If ß d (N) denotes the number of pairs less than N , then ß d (N) ¸ 2c 2 N log 2 (N) Y pjd p \Gamma 1 p \Gamma 2 : (1) Here the constant c 2 is defined in the following way 1 : c 2 j Y p?2 ` 1 \Gamma 1 (p \Gamma 1) 2 ' = 0:66016 : : : (2) The computer results of the search for pairs of primes separated by a distance d and smaller than N for N = 2 22 ; 2 24 ; : : : ; 2 40 ß 1:1 \Theta 10 12 are shown in the Fig.1. The characteristic oscillating pattern of points is caused by the product J(d) = Y pjd;p?2 p \Gamm...
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
First Occurrence of a Given Gap between Consecutive Primes
, 1997
"... Heuristic arguments are given, that the pair of consecutive primes separated by a distance d appears for the first time at p f (d) ¸ p d exp ` 1 2 q ln 2 (d) + 4d ' . The comparison with the results of the computer search provides the support for the conjectured formula. ..."
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Heuristic arguments are given, that the pair of consecutive primes separated by a distance d appears for the first time at p f (d) ¸ p d exp ` 1 2 q ln 2 (d) + 4d ' . The comparison with the results of the computer search provides the support for the conjectured formula.
Characterization of the distribution of twin primes
 In: arXiv:math.NT/0103191
"... We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows ..."
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We adopt an empirical approach to the characterization of the distribution of twin primes within the set of primes, rather than in the set of all natural numbers. The occurrences of twin primes in any finite sequence of primes are like fixed probability random events. As the sequence of primes grows, the probability decreases as the reciprocal of the count of primes to that point. The manner of the decrease is consistent with the Hardy–Littlewood Conjecture, the Prime Number Theorem, and the Twin Prime Conjecture. Furthermore, our probabilistic model, is simply parameterized. We discuss a simple test which indicates the consistency of the model extrapolated outside of the range in which it was constructed.
Implications of a New Characterisation of the Distribution of Twin Primes
, 2001
"... We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the “high jumper ” i.e., the separat ..."
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We bring to bear an empirical model of the distribution of twin primes and produce two distinct results. The first is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the “high jumper ” i.e., the separation with greatest likelihood (in terms of primes) is always expected to be zero.
JUMPING CHAMPIONS AND GAPS BETWEEN CONSECUTIVE PRIMES
, 910
"... Abstract. For any real x, the most common difference that occurs among the consecutive primes less than or equal to x is called a jumping champion. This term was introduced by J. H. Conway in 1993. There are occasionally ties. Therefore there can be more than one jumping champion for a given x. The ..."
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Abstract. For any real x, the most common difference that occurs among the consecutive primes less than or equal to x is called a jumping champion. This term was introduced by J. H. Conway in 1993. There are occasionally ties. Therefore there can be more than one jumping champion for a given x. The first, but shortlived, jumping champion is 1. Aside from the numerical studies, nothing else has been proved for other jumping champions as x increases. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf formulated, on the basis of heuristic and empirical evidence, the conjecture that the numbers greater than 1 that are jumping champions are 4 and the sequence of primorials 2, 6, 30, 210, 2310,.... The authors pointed out that this conjecture is not a direct consequence of other deep conjectures concerning primes. Therefore they made a weaker and possibly more accessible conjecture, that any fixed prime p divides all sufficiently large jumping champions. In the present paper we shall extend the work of P. Erdős and E. G. Straus from 1980 to prove that this second conjecture follows directly from the prime pair conjecture of G. H. Hardy and J. E. Littlewood. 1. The most likely differences between consecutive primes In an issue of the 1977–78 volume of the Journal of Recreational Mathematics, H. Nelson [5] proposed the following unsolved Problem 654:
DRAFT KRMK Implications of a New Characterisation of the Distribution of Twin Primes
, 2001
"... We bring to bear an empirical model of the distribution of twin primes and produce three distinct results pertinent to twins and, by extension, evidence against the Riemann Hypothesis. The first result is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the seque ..."
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We bring to bear an empirical model of the distribution of twin primes and produce three distinct results pertinent to twins and, by extension, evidence against the Riemann Hypothesis. The first result is that we can make a quantitative probabilistic prediction of the occurrence of gaps in the sequence of twins within the primes. The second is that the “high jumper ” i.e., the separation with greatest likelihood (in terms of primes) is always expected to be zero. The third is an elementary proof that Brun’s constant is bounded, i.e., that the series of reciprocal twins converges. We will demonstrate that our elementary proof is necessarily flawed because it is too strong, and attribute its failure to the fact that the error term was neglected in the model for the distribution of the primes. It is made very clear that the proof is incorrect by the fact that it is easily adapted to demonstrate that sums of subsequences of reciprocal primes are bounded, whereas it is clear that all series of the type we consider are in fact divergent. Attempts to explicitly model the behaviour of the error term require consideration of the
Query Access Assurance in Outsourced Databases
 IEEE TRANSACTIONS ON SERVICES COMPUTING
"... Query execution assurance is an important concept in defeating lazy servers in the database as a service model. We show that extending query execution assurance to outsourced databases with multiple data owners is highly inefficient. To cope with lazy servers in the distributed setting, we propose q ..."
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Query execution assurance is an important concept in defeating lazy servers in the database as a service model. We show that extending query execution assurance to outsourced databases with multiple data owners is highly inefficient. To cope with lazy servers in the distributed setting, we propose query access assurance (QAA) that focuses on IObound queries. The goal in QAA is to enable clients to verify that the server has honestly accessed all records that are necessary to compute the correct query answer, thus eliminating the incentives for the server to be lazy if the query cost is dominated by the IO cost in accessing these records. We formalize this concept for distributed databases, and present two efficient schemes that achieve QAA with high success probabilities. The first scheme is simple to implement and deploy, but may incur excessive server to client communication cost and verification cost at the client side, when the query selectivity or the database size increases. The second scheme is more involved, but successfully addresses the limitation of the first scheme. Our design employs a few number theory techniques. Extensive experiments demonstrate the efficiency, effectiveness and usefulness of our schemes.
IEEE TRANSACTIONS ON SERVICES COMPUTING 1 Query Access Assurance in Outsourced Databases
"... Abstract—Query execution assurance is an important concept in defeating lazy servers in the database as a service model. We show that extending query execution assurance to outsourced databases with multiple data owners is highly inefficient. To cope with lazy servers in the distributed setting, we ..."
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Abstract—Query execution assurance is an important concept in defeating lazy servers in the database as a service model. We show that extending query execution assurance to outsourced databases with multiple data owners is highly inefficient. To cope with lazy servers in the distributed setting, we propose query access assurance (QAA) that focuses on IObound queries. The goal in QAA is to enable clients to verify that the server has honestly accessed all records that are necessary to compute the correct query answer, thus eliminating the incentives for the server to be lazy if the query cost is dominated by the IO cost in accessing these records. We formalize this concept for distributed databases, and present two efficient schemes that achieve QAA with high success probabilities. The first scheme is simple to implement and deploy, but may incur excessive server to client communication cost and verification cost at the client side, when the query selectivity or the database size increases. The second scheme is more involved, but successfully addresses the limitation of the first scheme. Our design employs a few number theory techniques. Extensive experiments demonstrate the efficiency, effectiveness and usefulness of our schemes. Index Terms—Database as a service, database security, quality of services, query assurance, service enforcement and assurance. 1