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34
Ramanujan’s Lost Notebook I
, 2005
"... University. Dr. Lucy Slater had suggested to him that there were materials deposited there from the estate of the late G.N. Watson that might be of interest to him. In ..."
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Cited by 27 (4 self)
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University. Dr. Lucy Slater had suggested to him that there were materials deposited there from the estate of the late G.N. Watson that might be of interest to him. In
Fuzzycast: Efficient VideoonDemand over Multicast
 in Proceedings of Infocom 2002
, 2002
"... Server bandwidth has been identified as a major bottleneck in large VideoonDemand (VoD) systems. Using multicast delivery to serve popular content helps increase scalability by making efficient use of server bandwidth. In addition, recent research has focused on proactive schemes in which the serv ..."
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Cited by 17 (7 self)
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Server bandwidth has been identified as a major bottleneck in large VideoonDemand (VoD) systems. Using multicast delivery to serve popular content helps increase scalability by making efficient use of server bandwidth. In addition, recent research has focused on proactive schemes in which the server periodically multicasts popular content without explicit requests from clients. Proactive schemes are attractive because they consume bounded server bandwidth irrespective of client arrival rate. In this work, we describe Fuzzycast, a scalable periodic multicast scheme that uses simple techniques to provide video on demand at reasonable client startup times while consuming optimal server bandwidth. We present a theoretical analysis of its bandwidth and client buffer requirements and prove its optimality. We study the effect of variable bitrate (VBR) media on Fuzzycast performance and propose a simple extension to transmit VBR media over constant rate channels. Finally, we solve the problem of partitioning a transmission over multiple multicast groups by considering it as a specific instance of a more widely encountered resource tradeoff.
Nagaraj, Density of Carmichael numbers with three prime factors
 Math.Comp.66 (1997), 1705–1708. MR 98d:11110
"... Abstract. We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors. 1. ..."
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Cited by 10 (0 self)
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Abstract. We get an upper bound of O(x 5/14+o(1) ) on the number of Carmichael numbers ≤ x with exactly three prime factors. 1.
NonAbelian Generalizations of the ErdősKac Theorem
, 2001
"... Abstract. Let a be a natural number greater than 1. Let fa(n) be the order of a mod n. Denote by ω(n) the number of distinct prime factors of n. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance: The number of n ≤ x coprime to a sati ..."
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Cited by 7 (5 self)
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Abstract. Let a be a natural number greater than 1. Let fa(n) be the order of a mod n. Denote by ω(n) the number of distinct prime factors of n. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance: The number of n ≤ x coprime to a satisfying
COUNTING CONGRUENCE SUBGROUPS
"... Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence ..."
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Cited by 4 (2 self)
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Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence
Counting the number of solutions to the ErdősStraus equation on unit fractions
"... Abstract. For any positive integer n, let f(n) denote the number of solutions to the Diophantine equation 4 1 1 1 = + + with x, y, z positive integers. The ErdősStraus conjecture asserts that n x y z f(n)> 0 for every n � 2. To solve this conjecture, it suffices without loss of generality to con ..."
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Abstract. For any positive integer n, let f(n) denote the number of solutions to the Diophantine equation 4 1 1 1 = + + with x, y, z positive integers. The ErdősStraus conjecture asserts that n x y z f(n)> 0 for every n � 2. To solve this conjecture, it suffices without loss of generality to consider the case when n is a prime p. In this paper we consider the question of bounding the sum ∑ p<N f(p) asymptotically as N → ∞, where p ranges over primes. Our main result establishes the asymptotic upper and lower bounds N log 2 N ≪ ∑ f(p) ≪ N log 2 N log log N. p�N In particular, f(p) = Oδ(log3 p log log p) for a subset of primes of density δ arbitrarily close to 1. Also, for a subset of the primes with density 1 the following lower bound holds: f(p) ≫ (log p) 0.549. These upper and lower bounds show that a typical prime has a small number of solutions to the ErdősStraus Diophantine equation; small, when compared with other additive problems, like Waring’s problem. We establish several more results on f and related quantities, for instance the bound f(p) ≪ p 3 5 +O ( 1 log log p) for all primes p. Eventually we prove lower bounds for the number fm,k(n) of solutions of m n
Building Pseudoprimes With A Large Number Of Prime Factors
, 1995
"... We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong ..."
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Cited by 2 (0 self)
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We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.
A new algorithm for constructing large Carmichael
 Ken Nakamula, Department of Mathematics and Information Sciences, Tokyo Metropolitan University, MinamiOsawa, Hachioji
, 1996
"... Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 11015 ..."
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Abstract. We describe an algorithm for constructing Carmichael numbers N with a large number of prime factors p1,p2,...,pk. This algorithm starts with a given number Λ = lcm(p1 − 1,p2 −1,...,pk − 1), representing the value of the Carmichael function λ(N). We found Carmichael numbers with up to 1101518 factors. 1.