300mm prime gaps that need to be addressed to boost productivity

The Firoozbakht, Nicholson, and Farhadian conjectures can be phrased in terms of increasingly powerful conjectured bounds on the prime gaps g n : = p n + 1 - p n . While a general proof of any of these conjectures is far out of reach, I shall show that all three of these conjectures are unconditionally and explicitly verified for all primes below the as yet unknown location of the 81st maximal prime gap, certainly for all primes p < 2 64 . For the Firoozbakht conjecture itself this is a rather minor improvement on currently known results, but for the somewhat stronger Nicholson and Farhadian conjectures this may be considerably more interesting. Sequences: A005250 A002386 A005669 A000101 A107578 A246777 A246776.

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300-mm Prime Gaps That Need to Be Addressed to Boost Productivity

IEEE Transactions on Semiconductor Manufacturing ◽

10.1109/tsm.2008.2005363 ◽

2008 ◽

Vol 21 (4) ◽

pp. 592-599 ◽

Cited By ~ 3

Author(s):

Les Marshall ◽

Kristin Rust ◽

Kilian Schmidt

Keyword(s):

Prime Gaps

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The Maximal Prime Gaps Supremum and the Firoozbakht's Hypothesis No 30

10.20944/preprints202006.0366.v1 ◽

2020 ◽

Author(s):

Jan Feliksiak

Keyword(s):

Upper Bound ◽

Prime Numbers ◽

Heuristic Argument ◽

Mathematical Problems ◽

The Subject ◽

Prime Gaps ◽

Theoretical Results

The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of the current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a researchin the subject of the maximal prime gaps. the document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht's Hypothesis No 30. Firoozbakht's Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously, the Firoozbakht's Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht.

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The Elementary Proof of the Riemann's Hypothesis

10.20944/preprints202006.0365.v1 ◽

2020 ◽

Author(s):

Jan Feliksiak

Keyword(s):

Mathematical Theory ◽

Elementary Proof ◽

Counting Function ◽

Binomial Coefficients ◽

Complex Issue ◽

Logarithmic Integral ◽

Prime Counting Function ◽

Prime Gaps ◽

The Relationship ◽

Very High

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

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The Elementary Proof of the Riemann's Hypothesis

10.20944/preprints202006.0365.v2 ◽

2021 ◽

Author(s):

Jan Feliksiak

Keyword(s):

Mathematical Theory ◽

Elementary Proof ◽

Counting Function ◽

Binomial Coefficients ◽

Complex Issue ◽

Logarithmic Integral ◽

Prime Counting Function ◽

Prime Gaps ◽

The Relationship ◽

Very High

This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function Pi(n). Due to its very high precision, it permits to verify the relationship between the prime counting function Pi(n) and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the RIemann's Hypothesis conclusively.

Download Full-text