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Fast Generation of Prime Numbers and Secure PublicKey Cryptographic Parameters
, 1995
"... A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. The ..."
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A very efficient recursive algorithm for generating nearly random provable primes is presented. The expected time for generating a prime is only slightly greater than the expected time required for generating a pseudoprime of the same size that passes the MillerRabin test for only one base. Therefore our algorithm is even faster than presentlyused algorithms for generating only pseudoprimes because several MillerRabin tests with independent bases must be applied for achieving a sufficient confidence level. Heuristic arguments suggest that the generated primes are close to uniformly distributed over the set of primes in the specified interval. Security constraints on the prime parameters of certain cryptographic systems are discussed, and in particular a detailed analysis of the iterated encryption attack on the RSA publickey cryptosystem is presented. The prime generation algorithm can easily be modified to generate nearly random primes or RSAmoduli that satisfy t...
On elementary proofs of the Prime Number Theorem for arithmetic progressions, without characters.
, 1993
"... : We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a s ..."
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: We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that `(x; q; a) ¸ 2x=OE(q) for each a 62 H and `(x; q; a) = o(x=OE(q)), otherwise. From here, we deduce that if the second case holds at all, then it holds only for the multiples of some fixed integer q 0 ? 1. Actually, even if the Prime Number Theorem for arithmetic progressions, modulo q, does hold, these methods allow us to deduce the behaviour of a possible `Siegel zero' from Selberg's formula. We also propose a new method for determining explicit upper and lower bounds on `(x; q; a), which uses only elementary number theoretic computations. 1. Introduction. Define `(x) = P px log p, where p only denot...
The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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Cited by 4 (4 self)
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.
RamanujanFourier series, the WienerKhintchine formula and the distribution of prime pairs
, 1999
"... The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PA ..."
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Cited by 4 (2 self)
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The WienerKhintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a WienerKhintchine formula. "Experimental" evidence is given for this. c # 1999 Elsevier Science B.V. All rights reserved. PACS: 05.40+j; 02.30.Nw; 02.10.Lh Keywords: Twin primes; RamanujanFourier series; WienerKhintchine formula 1. Introduction " The WienerKhintchine theorem states a relationship between two important characteristics of a random process: the power spectrum of the process and the correlation function of the process" [1]. One of the outstanding problems in number theory is the problem of prime pairs which asks how primes of the form p and p+h (where h is an even integer) are distributed. One immediately notes that this is a problem of #nding correlation between primes. We make two key observations. First of all there is an arithmetical function (a function de#ned on integers) which traps the...
Partitions of Planar Sets Into Small Triangles
, 1985
"... Given 3n points in the unit square, n ³ 2, they determine n triangles whose vertices exhaust the given 3n points in many ways. Choose the n triangles so that the sum of their areas is minimal, and let a*(n) be the maximum value of this minimum over all configurations of 3n points. Then n  1/2 << ..."
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Given 3n points in the unit square, n ³ 2, they determine n triangles whose vertices exhaust the given 3n points in many ways. Choose the n triangles so that the sum of their areas is minimal, and let a*(n) be the maximum value of this minimum over all configurations of 3n points. Then n  1/2 << a*(n) << n  1/9 is deduced using results on the Heilbronn triangle problem. If the triangles are required to be area disjoint it is not even clear that the sum of their areas tends to zero; this open question is examined in a slightly more general setting. 1980 Mathematics Subject Classification: Primary 52A37, 52A40. Secondary 52A45. Key Words and Phrases: Convex body, convex hull, disjoint triangulation, Heilbronn's problem, triangulation. Partitions of planar sets into small triangles by Andrew M. Odlyzko AT&T Bell Laboratories Murray Hill, NJ 07974 Janos Pintz Mathematical Institute Hungarian Academy of Sciences H1053 Budapest Realtanoda u. 1315 Hungary and Kenneth B. Stola...
The approximation
"... The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x>1, let π(x) denote the number of primes less ..."
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The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number x>1, let π(x) denote the number of primes less than x. The prime number theorem is the assertion that lim x→ ∞ π(x) x
Some InformationTheoretic Computations . . .
 SUBMITTED TO JORMA RISSANEN’S FESTSCHRIFT VOLUME
, 2008
"... We illustrate how elementary informationtheoretic ideas may be employed to provide proofs for wellknown, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all prim ..."
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We illustrate how elementary informationtheoretic ideas may be employed to provide proofs for wellknown, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all primes p not exceeding n. We also give finiten bounds refining the above limit. This result, originally proved by Chebyshev in 1852, is closely related to the celebrated prime number theorem.
Prime numbers
"... · · · prime numbers “grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. ” Don Zagier Abstract. In my talk I will pose several questions about prime numbers. We will see that on the one hand some of the ..."
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· · · prime numbers “grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. ” Don Zagier Abstract. In my talk I will pose several questions about prime numbers. We will see that on the one hand some of them allow an answer with a proof of just a few lines, on the other hand, some of them lead to deep questions and conjectures not yet understood. This seems to represent a general pattern in mathematics: your curiosity leads to a study of “easy ” questions related with quite deep structures. I will give examples, suggestions and references for further study. This elementary talk was meant for freshman students; it is not an introduction to number theory, but it can be considered as an introduction: “what is mathematics about, and how can you enjoy the fascination of questions and insights?”
THE WIENER–IKEHARA THEOREM BY COMPLEX ANALYSIS JAAP KOREVAAR
"... Abstract. The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman’s contour integration method can be adapted to establish the Wiener–Ikehara theorem. A simple special case suffices for the PNT. But what about the twinprime p ..."
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Abstract. The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman’s contour integration method can be adapted to establish the Wiener–Ikehara theorem. A simple special case suffices for the PNT. But what about the twinprime problem? 1.
An Epic Drama: The Development of the Prime Number Theorem
"... Abstract. The prime number theorem, describing the aymptotic density of the prime numbers, has often been touted as the most surprising result in mathematics. The statement and development of the theorem by Legendre, Gauss and others and its eventual proof by Hadamard and de al ValléePoussin span t ..."
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Abstract. The prime number theorem, describing the aymptotic density of the prime numbers, has often been touted as the most surprising result in mathematics. The statement and development of the theorem by Legendre, Gauss and others and its eventual proof by Hadamard and de al ValléePoussin span the whole nineteenth century and encompass the growth of a brand new field in analytic number theory. As an outgrowth of the techniques of the proof is the Riemann hypothesis which today is perhaps the outstanding open problem in mathematics. These ideas and occurences certainly constitute an epic drama within the history of mathematics and one that is not as well known among the general mathematical community as it should be. In the present paper we trace out the paper, the development of the proof and a raft of other ideas, results and concepts that come from the prime number theorem.