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



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          This paper addresses the concept of maximal prime gaps. The established upper bound for the maximal prime gaps becomes in fact the Supremum bound. The Supremum bound is crucial in understanding the prime numbers distribution. It will prove to be a key element in verification of many mathematical theorems, aiding to supplant often implemented heuristics. The first implementation of the Supremum bound is to verify the veracity of the conjecture made by an Iranian mathematician Farideh Firoozbakht as well as both prime gaps bounds implied by the conjecture. 


          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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          MDPI AG
          June 30 2020
          © 2020

          The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

          Number theory, Mathematics

          prime gaps, maximal prime gaps Supremum, Legendre conjecture, Farideh Firoozbakht, Firoozbakht conjecture, elementary proof, Cramer's conjecture


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