6 articles for “Prime numbers”

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Research ArticleOpen accessPhrase in the titleA Simple and Accurate Relation Between the Logarithm Integral Li(x) and the Primes Counting Function π(x) is Derived Making use of the O.E.I.S. Prime Numbers “Sequences”Paolo AscarelliToday the prime numbers π(x) contained under the number x appears to be somewhat overestimated by the logarithm integral function Li(x) and underestimated by the function x/ln(x), both originally proposed by Gauss around 1792-1796. However, a simple and accurate expression, relating Li(x) and π(x), may be derived using the data reported on the O.E.I.S. “Sequences”. This relation can also suggest … …Journal of Mathematics and Statistics, Vol. 17, Issue 1 · 30 Sep 2021 · 7,739 accessesView PDF Cite

Research ArticleOpen accessPhrase in the titleThe Linear and Quadratic Nature of Sequences for Determining Prime Numbers by Elimination of CompositesJonathan M. Dugas, Brian M. O’ConnorA previously derived set of sequences for generating prime numbers is modified by various algebraic and trigonometric manipulations. The modified sequences are then used to more efficiently generate prime numbers. The modified sequences are simplified from their original form and are much easier to implement into a computer program. It is shown by computation that the modified sequences more efficiently … …Journal of Mathematics and Statistics, Vol. 14, Issue 1 · 16 May 2018 · 7,405 accessesView PDF Cite

HypothesisOpen accessPhrase in the titleSequences for Determination of Prime Numbers by Elimination of CompositesJonathan M. Dugas, Brian M. O’ConnorIn this study, sequences are used for direct computation of prime numbers. A single equation for generating all prime numbers with the exception of 2 and 3 and composite numbers that are not divisible by 2 or 3 is presented. A periodicity to the indices of the composites generated by that equation is determined. An equation to determine the indices … …Journal of Mathematics and Statistics, Vol. 13, Issue 3 · 20 Jul 2017 · 8,042 accessesView PDF Cite

Research ArticleOpen accessPhrase in the titleEfficient Realization of S-Box based reduced Residue of Prime Numbers using Virtex-5 and Virtex-6 FPGAsMohammed H. Al MijalliProblem statement: The S-Box transformation is a computationally intensive and important operation of the Advanced Encryption Standard (AES). Approach: This study presents the comparative study between reduced Residue of Prime Numbers and Galois Filed GF (28) based SBoxes using Virtex-5 and Virtex-6 FPGA devices. The implementation of S-Boxes is done using Very High speed integrated circuit Hardware Description Language (VHDL). … …American Journal of Applied Sciences, Vol. 8, Issue 8 · 18 Jul 2011 · 6,975 accessesView PDF Cite

Research ArticleOpen accessEvery word in the titleThe Importance of Numbers and the Need to Study Primes: The Prime QuestionsMatthew Curtis, Gurudeo A. TularamProblem statement: The study of prime numbers is important today as it was in ancient times. The level of interest in any form of mathematics of students in Australian high schools is decreasing and furthermore the level of interest in number theory is also less that it was before; prime numbers is not being considered as an area of interest … …Journal of Mathematics and Statistics, Vol. 7, Issue 4 · 14 Sep 2011 · 9,656 accessesView PDF Cite

Research ArticleOpen accessIn the abstractA Numerical Test on the Riemann Hypothesis with ApplicationsN. K. Oladejo, I. A. AdetundeProblem statement: The Riemann hypothesis involves two products of the zeta function ζ(s) which are: Prime numbers and the zeros of the zeta function ζ(s). It states that the zeros of a certain complex-valued function ζ (s) of a complex number s ≠ 1 all have a special form, which may be trivial or non trivial. Zeros at the negative … …Journal of Mathematics and Statistics, Vol. 5, Issue 1 · 31 Mar 2009 · 7,859 accessesView PDF Cite