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Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers 被引量:1
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作者 Ramon Carbó-Dorca 《Applied Mathematics》 2022年第6期538-543,共6页
A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the preci... A simple recursive algorithm to generate the set of natural numbers, based on Mersenne numbers: M<sub>N</sub> = 2<sup>N</sup> – 1, is used to count the number of prime numbers within the precise Mersenne natural number intervals: [0;M<sub>N</sub>]. This permits the formulation of an extended twin prime conjecture. Moreover, it is found that the prime numbers subsets contained in Mersenne intervals have cardinalities strongly correlated with the corresponding Mersenne numbers. 展开更多
关键词 mersenne numbers Recursive Generation of Natural numbers mersenne Natural Number Intervals Counting the Number of Prime numbers in mersenne Natural Intervals Correlation between Prime Number Set Cardinalities and mersenne numbers Extended Twin Prime Number Conjecture
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Elliptic Curve Point Multiplication by Generalized Mersenne Numbers 被引量:2
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作者 Tao Wu Li-Tian Liu 《Journal of Electronic Science and Technology》 CAS 2012年第3期199-208,共10页
Montgomery modular multiplication in the residue number system (RNS) can be applied for elliptic curve cryptography. In this work, unified modular multipliers over generalized Mersenne numbers are proposed for RNS M... Montgomery modular multiplication in the residue number system (RNS) can be applied for elliptic curve cryptography. In this work, unified modular multipliers over generalized Mersenne numbers are proposed for RNS Montgomery modular multiplication, which enables efficient elliptic curve point multiplication (ECPM). Meanwhile, the elliptic curve arithmetic with ECPM is performed by mixed coordinates and adjusted for hardware implementation. In addition, the conversion between RNS and the binary number system is also discussed. Compared with the results in the literature, our hardware architecture for ECPM demonstrates high performance. A 256-bit ECPM in Xilinx XC2VP100 field programmable gate array device (FPGA) can be performed in 1.44 ms, costing 22147 slices, 45 dedicated multipliers, and 8.25K bits of random access memories (RAMs). 展开更多
关键词 Elliptic curve cryptography generalized mersenne numbers modular multiplier residue number system.
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Origin of Sexy Prime Numbers, Origin of Cousin Prime Numbers, Equations from Supposedly Prime Numbers, Origin of the Mersenne Number, Origin of the Fermat Number
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作者 Mady Ndiaye 《Advances in Pure Mathematics》 2024年第5期321-332,共12页
We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their ... We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their origin then we classified them into two categories according to their classes, we showed in which context two prime numbers which differ from 6 are called sexy and in what context they are said real sexy prime. For those whose difference is equal to 4, we showed their origin then we showed that two prime numbers which differ from 4, that is to say two cousin prime numbers, are successive. We made an observation on the supposed prime numbers then we established two pairs of equations from this observation and deduced the origin of the Mersenne number and that of the Fermat number. 展开更多
关键词 Cousin Prime numbers Sexy Prime numbers Real Sexy Prime numbers Equations from Supposed Prime numbers mersenne Number Fermat Number Supposed Prime numbers Prime numbers
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