Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

Silva, Tomás Oliveira e; Herzog, S.; Pardi, S.

doi:10.1090/s0025-5718-2013-02787-1

Cited by 114 publications

(118 citation statements)

References 44 publications

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“…Again almost identically, we have * [8, 4 · 10 18 ] is a sum of two distinct primes p and q (see [31]). Hence, 5 is the only odd nonaliquot number up to 10…”

Section: Paul Pollack and Carl Pomerancementioning

confidence: 99%

Some problems of Erdős on the sum-of-divisors function

Pollack

Pomerance

2016

Trans. Amer. Math. Soc. Ser. B

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Abstract. Let σ(n) denote the sum of all of the positive divisors of n, and let s(n) = σ(n) − n denote the sum of the proper divisors of n. The functions σ(·) and s(·) were favorite subjects of investigation by the late Paul Erdős. Here we revisit three themes from Erdős's work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n not in the range of the function s(·); these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k-sets, where a friendly k-set is a collection of k distinct integers which share the same value of σ(n) n .

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“…Again almost identically, we have * [8, 4 · 10 18 ] is a sum of two distinct primes p and q (see [31]). Hence, 5 is the only odd nonaliquot number up to 10…”

Section: Paul Pollack and Carl Pomerancementioning

confidence: 99%

Some problems of Erdős on the sum-of-divisors function

Pollack

Pomerance

2016

Trans. Amer. Math. Soc. Ser. B

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“…In general, we can decompose the total number of nodes into numerous prime‐sized sets and add a handful of dummy nodes or have some nodes overlap as needed. Also note that Goldbach is proved up to 4·10 18 , so it is not a conjecture for our practical interest, as shown in Oliveira et al…”

Section: Topology Manipulationmentioning

confidence: 85%

Managing risk in high assurance systems by optimizing topological resources

Hyden

McGrath

Moskowitz

et al. 2017

J Software Evolu Process

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We explore strategies for manipulating the topology of a network to promote increased and pragmatic high assurance systems. Topology matters to network threats and security, and the relative distance between nodes can impact the rate of dispersion of viruses, as well as access times in denial of service, probing, and insider threat attacks. We suggest methods to separate threatening and threatened nodes with enough hops to reduce and degrade risks. This work provides network analysts with an option to include other measures such as risk assessments to the construction and management of high assurance systems. We consider a scaled down model to demonstrate the proof of concept using artificial data. Specifically, we explore the efficacy of ring networks and the structure that occurs on k‐hop networks when there are a prime number of nodes. We introduce techniques for the randomization of network topologies to manage real‐time risk and provide a dynamic means to improve network security by increasing technical debt to a potential attacker.

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“…It so remains to prove this result for all integers in the range 3 ≤ n < 10 10 . If n is even, we have the numerical verification by Oliviera e Silva, Herzog and Pardi [7] that all even integers up to 4 · 10 18 can be written as the sum of two primes. Thus, every even integer greater than two may be written as the sum of a prime and a square-free number.…”

Section: A Lower Bound For R(n)mentioning

confidence: 92%