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
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
Mersenne primes are a special kind of primes, which are always an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. It has not settled that whether the...Mersenne primes are a special kind of primes, which are always an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. It has not settled that whether there exist infinite Mersenne primes. And several of conjectures on the distribution of it provided by scholars. Starting from the Mersenne primes known about, in this paper we study the distribution of Mersenne primes and argued against some suppositions by data analyzing.展开更多
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
With the rapid development of digital information technology,images are increasingly used in various fields.To ensure the security of image data,prevent unauthorized tampering and leakage,maintain personal privacy,and...With the rapid development of digital information technology,images are increasingly used in various fields.To ensure the security of image data,prevent unauthorized tampering and leakage,maintain personal privacy,and protect intellectual property rights,this study proposes an innovative color image encryption algorithm.Initially,the Mersenne Twister algorithm is utilized to generate high-quality pseudo-random numbers,establishing a robust basis for subsequent operations.Subsequently,two distinct chaotic systems,the autonomous non-Hamiltonian chaotic system and the tentlogistic-cosine chaotic mapping,are employed to produce chaotic random sequences.These chaotic sequences are used to control the encoding and decoding process of the DNA,effectively scrambling the image pixels.Furthermore,the complexity of the encryption process is enhanced through improved Joseph block scrambling.Thorough experimental verification,research,and analysis,the average value of the information entropy test data reaches as high as 7.999.Additionally,the average value of the number of pixels change rate(NPCR)test data is 99.6101%,which closely approaches the ideal value of 99.6094%.This algorithm not only guarantees image quality but also substantially raises the difficulty of decryption.展开更多
Mersenne primes are a special kind of primes, which are an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. Searching for Mersenne primes is very chal...Mersenne primes are a special kind of primes, which are an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. Searching for Mersenne primes is very challenging in scientific researches. In this paper, the numbers of thousand place of Mersenne primes are studied, and the conclusion is presented by using the Chinese remainder theorem.展开更多
Digital roots of numbers have several interesting properties, most of which are well-known. In this paper, our goal is to prove some lesser known results concerning the digital roots of powers of numbers in an arithme...Digital roots of numbers have several interesting properties, most of which are well-known. In this paper, our goal is to prove some lesser known results concerning the digital roots of powers of numbers in an arithmetic progression. We will also state some theorems concerning the digital roots of Fermat numbers and star numbers. We will conclude our paper by an interesting application.展开更多
By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or ...By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.展开更多
The sum of reciprocals of Mersenne primes converges to 0.51645417894078856533···, which is an example of a probably infinite subset of primes whose sum of reciprocals is finite and can be computed accur...The sum of reciprocals of Mersenne primes converges to 0.51645417894078856533···, which is an example of a probably infinite subset of primes whose sum of reciprocals is finite and can be computed accurately. This value is larger than , where ?is the set of perfect powers of prime numbers.展开更多
文摘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.
基金supported by the National Natural Science Foundation of China under Grant No. 61073173
文摘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）.
文摘Mersenne primes are a special kind of primes, which are always an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. It has not settled that whether there exist infinite Mersenne primes. And several of conjectures on the distribution of it provided by scholars. Starting from the Mersenne primes known about, in this paper we study the distribution of Mersenne primes and argued against some suppositions by data analyzing.
文摘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.
基金supported by the Open Fund of Advanced Cryptography and System Security Key Laboratory of Sichuan Province(Grant No.SKLACSS-202208)the Natural Science Foundation of Chongqing(Grant No.CSTB2023NSCQLZX0139)the National Natural Science Foundation of China(Grant No.61772295).
文摘With the rapid development of digital information technology,images are increasingly used in various fields.To ensure the security of image data,prevent unauthorized tampering and leakage,maintain personal privacy,and protect intellectual property rights,this study proposes an innovative color image encryption algorithm.Initially,the Mersenne Twister algorithm is utilized to generate high-quality pseudo-random numbers,establishing a robust basis for subsequent operations.Subsequently,two distinct chaotic systems,the autonomous non-Hamiltonian chaotic system and the tentlogistic-cosine chaotic mapping,are employed to produce chaotic random sequences.These chaotic sequences are used to control the encoding and decoding process of the DNA,effectively scrambling the image pixels.Furthermore,the complexity of the encryption process is enhanced through improved Joseph block scrambling.Thorough experimental verification,research,and analysis,the average value of the information entropy test data reaches as high as 7.999.Additionally,the average value of the number of pixels change rate(NPCR)test data is 99.6101%,which closely approaches the ideal value of 99.6094%.This algorithm not only guarantees image quality but also substantially raises the difficulty of decryption.
文摘Mersenne primes are a special kind of primes, which are an important content in number theory. The study of Mersenne primes becomes one of hot topics of the nowadays science. Searching for Mersenne primes is very challenging in scientific researches. In this paper, the numbers of thousand place of Mersenne primes are studied, and the conclusion is presented by using the Chinese remainder theorem.
文摘Digital roots of numbers have several interesting properties, most of which are well-known. In this paper, our goal is to prove some lesser known results concerning the digital roots of powers of numbers in an arithmetic progression. We will also state some theorems concerning the digital roots of Fermat numbers and star numbers. We will conclude our paper by an interesting application.
文摘By extending both arithmetical operations into finite sets of natural numbers, from the entire set of natural numbers successively deleting some residue classes modulo a prime, we invented a recursive sieve method or algorithm on natural numbers and their sets. The algorithm mechanically yields a sequence of sets, which converges to the set of all primes p such that 2p + 1 divides the Mersenne number Mp. The cardinal sequence corresponding to the sequence of sets is strictly increasing. So that we have captured enough usable structures, without any estimation, the existing theories of those structures allow us to prove an exact result: there are infinitely many Mersenne composite numbers with prime exponents Mp.
文摘The sum of reciprocals of Mersenne primes converges to 0.51645417894078856533···, which is an example of a probably infinite subset of primes whose sum of reciprocals is finite and can be computed accurately. This value is larger than , where ?is the set of perfect powers of prime numbers.
