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Journal articles on the topic 'Prime number'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 November 2023

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1

Lichtman, Jared Duker, Greg Martin, and Carl Pomerance. "Primes in prime number races." Proceedings of the American Mathematical Society 147, no. 9 (June 14, 2019): 3743–57. http://dx.doi.org/10.1090/proc/14569.

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2

Chermidov, Sergey Ivanovich. "PRIME NUMBER LAW. DEPENDENCE OF PRIME NUMBERS ON THEIR ORDINAL NUMBERS AND GOLDBACH – EULER BINARY PROBLEM USING COMPUTER." Vestnik of Astrakhan State Technical University. Series: Management, computer science and informatics 2020, no. 4 (October 31, 2020): 80–100. http://dx.doi.org/10.24143/2072-9502-2020-4-80-100.

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The article considers the methods of defining and finding the distribution of composite numbers CN, prime numbers PN, twins of prime numbers Tw and twins of composite numbers TwCN that do not have divisors 2 and 3 in the set of natural numbers - ℕ based on a set of numbers like Θ = {6∙κ ± 1, κ ∈ ℕ}, which is a semigroup in relation to multiplication. There has been proposed a method of obtaining primes by using their ordinal numbers in the set of primes and vice versa, as well as a new algorithm for searching and distributing primes based on a closedness of the elements of the set Θ. It has been shown that a composite number can be presented in the form of products (6x ± 1) (6y ± 1), where x, y ℕ - are positive integer solutions of one of the 4 Diophantine equations: . It has been proved that if there is a parameter λ of prime twins, then none of Diophantine equations P (x, y, λ) = 0 has positive integer solutions. There has been found the new distribution law of prime numbers π(x) in the segment [1 ÷ N]. Any even number is comparable to one of the numbers i.e. . According to the above remainders m, even numbers are divided into 3 types, each type having its own way of representing sums of 2 elements of the set Θ. For any even number in a segment [1 ÷ ν], where ν = (ζ−m) / 6, , there is a parameter of an even number; it is proved that there is always a pair of numbers that are elements of the united sets of parameters of prime twins and parameters of transition numbers , i.e. numbers of the form with the same λ, if the form is a prime number, then the form is a composite number, and vice versa.

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3

Nakagoshi, Norikata. "On the class number of the lpth cyclotomic number field." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 2 (March 1991): 263–76. http://dx.doi.org/10.1017/s0305004100069735.

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The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick out some pairs (l, p) of two odd primes l and p such that the class number of each lpth cyclotomic number field is divisible by l even if p 1 (mod l). If p ≡ (mod l) for l ≥ 5 or p ≡ 1 (mod 32) for l = 3, then it is easy from the outset to achieve our intention of finding the factor l using the genus number formula. Otherwise it seems to be difficult. We wish to make it clear algebraically why the class number has the prime factor l.

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4

Farkas, Gábor, Zsombor Kiss, Dániel Papatyi, and Krisztina Schäffer. "Prime Numbers." Mérnöki és Informatikai Megoldások, no. II. (October 20, 2020): 5–13. http://dx.doi.org/10.37775/eis.2020.2.1.

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"``Prime hunting" can be considered as a research area of computational number theory. Its goal is to find special combinations of integers and prove their primality. Four research groups, established by A. Járai between 1992 and 2014, published numerous world class scientific results. In this period, due to Járai's arithmetic routines fastest in the world, they reached the world record 19 times, namely found the largest known twin primes 9 times, Sophie Germain primes 7 times, a prime of the form n^4+1, a number which is simultaneously twin and Sophie Germain prime and the three largest known primes forming a Cunningham chain of length 3 of the first kind. When A. Járai retired, the investigations of this area were suspended. In the beginning of 2020 the research was reopened by G. Farkas at the newly-founded campus (Szombathely) of ELTE. The first signal success came in the end of May 2020. They proved the primality of the numbers which form the largest known Cunningham chains of length 2 of the 2nd kind. In this paper we report on a newly started prime hunting project with the aim of increasing our students' research activity.

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5

Tallant, Jonathan. "Optimus prime: paraphrasing prime number talk." Synthese 190, no. 12 (May 29, 2011): 2065–83. http://dx.doi.org/10.1007/s11229-011-9959-8.

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6

Mackinnon, Nick. "Prime Number Formulae." Mathematical Gazette 71, no. 456 (June 1987): 113. http://dx.doi.org/10.2307/3616496.

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7

Ribenboim, Paulo. "Prime Number Records." College Mathematics Journal 25, no. 4 (September 1994): 280. http://dx.doi.org/10.2307/2687612.

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8

Gilbert Waltzek, Chris. "Prime Number Conjecture." Mathematics and Statistics 2, no. 6 (August 2014): 203–4. http://dx.doi.org/10.13189/ms.2014.020601.

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9

Granville, Andrew, and Greg Martin. "Prime Number Races." American Mathematical Monthly 113, no. 1 (January 1, 2006): 1. http://dx.doi.org/10.2307/27641834.

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10

Granville, Andrew, and Greg Martin. "Prime Number Races." American Mathematical Monthly 113, no. 1 (January 2006): 1–33. http://dx.doi.org/10.1080/00029890.2006.11920275.

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11

Granville, Andrew. "Prime Number Patterns." American Mathematical Monthly 115, no. 4 (April 2008): 279–96. http://dx.doi.org/10.1080/00029890.2008.11920529.

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12

Ribenboim, Paulo. "Prime Number Records." College Mathematics Journal 25, no. 4 (September 1994): 280–90. http://dx.doi.org/10.1080/07468342.1994.11973623.

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13

Latorre, Jose I., and German Sierra. "Quantum computation of prime number functions." Quantum Information and Computation 14, no. 7&8 (May 2014): 577–88. http://dx.doi.org/10.26421/qic14.7-8-3.

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We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than $2^n$, where $n$ is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.

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14

Appelqvist, Gunnar. "Squared prime numbers." JOURNAL OF ADVANCES IN MATHEMATICS 20 (February 14, 2021): 43–59. http://dx.doi.org/10.24297/jam.v20i.8952.

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My investigation shows that there is a regularity even by the prime numbers. This structure is obvious when a prime square is created. The squared prime numbers. 1. Connections in a prime square A prime square (or origin square) is defined as a square consisting of as many boxes as the origin prim squared. This prime settle every side of the square. So, for example, the origin square 17 have got four sides with 17 boxes along every side. The prime numbers in each of the 289 boxes are filled with primes when a prime number occur in the number series (1,2,3,4,5,6,7,8,9 and so on) and then is noted in that very box. If a box is occupied in the origin square A this prime number could be transferred to the corresponding box in a second square B, and thereafter the counting and noting continue in the first square A. Eventually we get two filled prime squares. Analyzing these squares, you leave out the right vertical line, representing only the origin prime number, When a square is filled with primes you subdivide it into four corner squares, as big as possible, denoted a, b, c and d clockwise. You also get a center line between the left and right vertical sides. Irrespective of what kind of constellation you activate this is what you find: Every constellation in the corner square a and/or d added to a corresponding constellation in the corner square b and/or c is evenly divisible with the origin prime. Every constellation in the corner square a and/or b added to a corresponding constellation in the corner square d and/or c is not evenly divisible with the origin prime. Every reflecting constellation inside two of the opposed diagonal corner squares, possibly summarized with any optional reflecting constellation inside the two other diagonal corner squares, is evenly divisible with the origin prime squared. You may even add a reflection inside the center line and get this result. My Conjecture 1 is that this applies to every prime square without end. A formula giving all prime numbers endless In the second prime square the prime numbers are always higher than in the first square if you compare a specific box. There is a mathematic connection between the prime numbers in the first and second square. This connection appears when you square and double the origin prime and thereafter add this number to the prime you investigate. A new higher prime is found after n additions. You start with lowest applicable prime number 3 and its square 3². Double it and you get 18. We add 18 to the six next prime numbers 5, 7, 11, 13, 17 and 19 in any order. After a few adds you get a prime and after another few adds you get another higher one. In this way you continue as long as you want to. The primes are creating themselves. A formula giving all prime numbers is: 5+18×n, +18×n, +18×n … without end 7+18×n, +18×n, +18×n … without end 11+18×n, +18×n, +18×n … without end 13+18×n, +18×n, +18×n … without end 17+18×n, +18×n, +18×n … without end 19+18×n, +18×n, +18×n … without end The letter n in the formula stands for how many 18-adds you must do until the next prime is found. My Conjecture 2 is that this you find every prime number by adding 18 to the primes 5, 7, 11, 13, 17 and 19 one by one endless. A method giving all prime numbers endless There is still a possibility to even more precise all prime numbers. You start a 5-number series derived from the start primes 7, 17, 19, 11, 13 and 5 in that very order. Factorized these number always begin with number 5. When each of these numbers are divided with five the quotient is either a prime number or a composite number containing of two or some more prime numbers in the nearby. By sorting out all the composite quotients you get all the prime numbers endless and in order. Every composite quotient starts with a prime from 5 and up, squared. Thereafter the quotients starting with that prime show up periodically according to a pattern of short and long sequences. The position for each new prime beginning the composite quotient is this prime squared and multiplied with 5. Thereafter the short sequence is this prime multiplied with 10, while the long sequence is this prime multiplied with 20. When all the composite quotients are deleted there are left several 5-numbers which divided with 5 give all prime numbers, and you even see clearly the distance between the prime numbers which for instance explain why the prime twins occur as they do. My Conjecture 3 is that this is an exact method giving all prime numbers endless and in order.

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15

Chenevier, Gaëtan. "On number fields with given ramification." Compositio Mathematica 143, no. 6 (November 2007): 1359–73. http://dx.doi.org/10.1112/s0010437x07003132.

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AbstractLet E be a CM number field and let S be a finite set of primes of E containing the primes dividing a given prime number l and another prime u split above the maximal totally real subfield of E. If ES denotes a maximal algebraic extension of E which is unramified outside S, we show that the natural maps $\mathrm {Gal}(\overline {E_u}/E_u) \longrightarrow \mathrm {Gal}(E_S/E)$ are injective. We discuss generalizations of this result.

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16

Annouk, Ikorong, and Paul Archambault. "A Short Note on Numbers of the Form K + Fn, Where K ? {2, 4, 8} and Fn is a Fermat Number." Sumerianz Journal of Scientific Research, no. 53 (August 26, 2022): 60–62. http://dx.doi.org/10.47752/sjsr.53.60.62.

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A Fermat number is a number of the form , where n is an integer 0. A Fermat composite (see Dickson [1] or Hardy and Wright [2] or Ikorong [3]) is a non-prime Fermat number and a Fermat prime is a prime Fermat number. Fermat composites and Fermat primes are characterized via divisibility in Ikorong [3] and in Ikorong [4]. It is known (see Ikorong [3]) that for every j ∈ {0, 1, 2, 3, 4}, Fj is a Fermat prime and it is also known (see Hardy and Wright [2] or Paul [5]) that F5 and F6 are Fermat composites 641×6700417, and since 2013, it is known that +1 is Fermat composite number). In this paper, we show [via elementary arithmetic congruences] the following Result (E.). For every integer n > 0 such that n ≡ 1 mod [2], we have Fn−1 ≡ 4 mod [7]; and for every integer n ≥ 2, we have Fn−1 ≡ 1 mod[j], where j ∈ {3, 5}. Result (E.) immediately implies that there are infinitely many composite numbers of the form 2 + Fn. Result (E.) also implies that the only prime of the form 4 + Fn is 7 and the only primes of the form 8+ Fn are twin primes 11 and 13. That being said, using result (E.) and a special case of a Theorem of Dirichlet on arithmetic progression, we conjecture that there are infinitely many primes of the form 2+Fn.

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17

Chillali, Abdelhakim. "R-prime numbers of degree k." Boletim da Sociedade Paranaense de Matemática 38, no. 2 (February 19, 2018): 75–82. http://dx.doi.org/10.5269/bspm.v38i2.38218.

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In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, "easy" and "hard" are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient of a function for it to be called one-way (see Theoretical Definition, in article). A twin prime is a prime number that has a prime gap of two, in other words, differs from another prime number by two, for example the twin prime pair (5,3). The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes p such that p + 2 is also prime. In this work we define a new notion: ‘r-prime number of degree k’ and we give a new RSA trap-door one-way. This notion generalized a twin prime numbers because the twin prime numbers are 2-prime numbers of degree 1.

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18

Ndiaye, Mady. "Prime Numbers." International Journal for Innovation Education and Research 5, no. 6 (June 30, 2017): 41–66. http://dx.doi.org/10.31686/ijier.vol5.iss6.672.

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A prime number is a natural number that has Just two divisors: one and itself. From antiquity until our time, scientists are researching mathematical reasoning to understand the prime numbers; eminent scholars had worked on this field before it is abandoned. Mathematicians considered the prime numbers like « building blocs in building natural numbers » and the field of mathematics the most difficult. Everything is about numbers, everything is about measure, The understanding of the natural numbers and more general the understanding of the numbers depend on the understanding of the prime numbers. This understanding of the prime will gives us greater ease to understand the other sciences. The prime numbers play a very important role for securing information technology hence promotion of the NTIC, Every year, there is a price for persons who will discover the biggest prime “it‟s the hunt for the big prime” This first part of this article about the prime numbers has taken a weight off the scientists „s shoulders by highlighting the universe of the prime numbers and has bring the problem of the prime numbers to an end. The mathematical formulas set out in this article allow us to determine all the biggest prime numbers compared to the capacity of our machines.

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19

Debnath, Lokenath, and Kanadpriya Basu. "Some Analytical and Computational Aspects of Prime Numbers, Prime Number Theorems and Distribution of Primes with Applications." International Journal of Applied and Computational Mathematics 1, no. 1 (November 25, 2014): 3–32. http://dx.doi.org/10.1007/s40819-014-0014-6.

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20

Andrian, Yudhi. "Perbandingan Penggunaan Bilangan Prima Aman Dan Tidak Aman Pada Proses Pembentukan Kunci." Creative Information Technology Journal 1, no. 3 (April 2, 2015): 194. http://dx.doi.org/10.24076/citec.2014v1i3.21.

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Algoritma ElGamal merupakan algoritma dalam kriptografi yang termasuk dalam kategori algoritma asimetris. Keamanan algoritma ElGamal terletak pada kesulitan penghitungan logaritma diskret pada bilangan modulo prima yang besar sehingga upaya untuk menyelesaikan masalah logaritma ini menjadi sangat sukar. Algoritma ElGamal terdiri dari tiga proses, yaitu proses pembentukan kunci, proses enkripsi dan proses dekripsi. Proses pembentukan kunci kriptografi ElGamal terdiri dari pembentukan kunci privat dan pembentukan kunci public. Pada proses ini dibutuhkan sebuah bilangan prima aman yang digunakan sebagai dasar pembentuk kunci public sedangkan sembarang bilangan acak digunakan sebagai pembentuk kunci privat. Pada penelitian sebelumnya digunakan bilangan prima aman pada proses pembentukan kunci namun tidak dijelaskan alasan mengapa harus menggunakan bilangan prima aman tersebut. Penelitian ini mencoba membandingkan penggunaan bilangan prima aman dan bilangan prima tidak aman pada pembentukan kunci algoritma elgamal. Analisa dilakukan dengan mengenkripsi dan dekripsi sebuah file dengan memvariasikan nilai bilangan prima aman dan bilangan prima tidak aman yang digunakan untuk pembentukan kunci public dan kunci privat. Dari hasil analisa dapat disimpulkan bahwa dengan menggunakan bilangan prima aman maupun bilangan prima tidak aman, proses pembentukan kunci, enkripsi dan dekripsi tetap dapat berjalan dengan baik, semakin besar nilai bilangan prima yang digunakan, maka kapasitas cipherteks juga semakin besar.Elgamal algorithm is an algorithm in cryptography that is included in the category of asymmetric algorithms. The security of Elgamal algorithm lies in the difficulty in calculating the discrete logarithm on large number of prime modulo that attempts to solve this logarithm problem becomes very difficult. Elgamal algorithm is consists of three processes, that are the key generating, encryption and decryption process. Key generation of elgamal cryptography process is consisted of the formation of the private key and public key. In this process requires a secure prime number is used as the basis for forming public key while any random number used as forming of the private key. In the previous research is used secure prime number on key generating process but does not explain the reasons of using the secure primes. This research tried to compare using secure and unsecure primes in elgamal key generating algorithm. The analysis is done by encrypting and decrypting a file by varying the value of secure and unsecure of prime numbers that are used on generating of a public and a private key. From the analysis it can be concluded that using secure and unsecure of prime numbers, the process of key generating, encryption and decryption can run well, the greater value of prime numbers are used, the greater the capacity of the ciphertext.

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Oktaviani, Dinni Rahma, Muhammad Habiburrohman, and Fiki Syaban Nugroho. "ALTERNATIVE PROOF OF THE INFINITUDE PRIMES AND PRIME PROPERTIES." BAREKENG: Jurnal Ilmu Matematika dan Terapan 17, no. 1 (April 20, 2023): 0475–80. http://dx.doi.org/10.30598/barekengvol17iss1pp0475-0480.

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Prime numbers is one of kind number that have many uses, one of which is cryptography. The uniqueness of prime numbers in their divisors and distributions causes prime numbers to be widely used in digital security systems. In number theory, one of famous theorem is Euclid theorem. Euclid theorem says about infinitely of prime numbers. Many alternative proof has been given by mathematician to find new theory or approximation of prime properties. The construction of proof give new idea about properties of prime number. So, in this study, we will give an alternative proof of Euclid theorem and investigate the properties of prime in distribution.

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22

KNOPFMACHER, ARNOLD, and FLORIAN LUCA. "ON PRIME-PERFECT NUMBERS." International Journal of Number Theory 07, no. 07 (November 2011): 1705–16. http://dx.doi.org/10.1142/s1793042111004447.

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We prove that the Diophantine equation [Formula: see text] has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization [Formula: see text], then [Formula: see text] represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.

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23

Chen, Y. G., and J. H. Fang. "On the number of weakly prime-additive numbers." Acta Mathematica Hungarica 160, no. 2 (October 24, 2019): 444–52. http://dx.doi.org/10.1007/s10474-019-01001-9.

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24

Carlos Guimaray Huerta, Héctor. "CARACTERIZACIÓN DE NÚMEROS PRIMOS." Revista Cientifica TECNIA 22, no. 2 (April 4, 2017): 25. http://dx.doi.org/10.21754/tecnia.v22i2.78.

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Los números primos es motivo de investigación en la teoría de números; en la actualidad, no existe una fórmula que nos permita obtener dichos números, y que la distribución de los mismos se considera que es aleatoria. Lo que existe son métodos para averiguar si un número es primo o compuesto. En el presente artículo se presenta una caracterización de números primos que es el complemento de los números compuestos. Palabras clave.- Divisor, Número primo, Número compuesto, Caracterización, Conjetura. ABSTRACTThe prime numbers motivate the investigation in number theory; nowadays, does not exist a formula that allows get those numbers, and that the distribution thereof is considered random. There are methods to find whether a number is prime or composite. This article presents a characterization of prime numbers which is the complement of composite numbers. Keywords.- Divisor, Prime number, Composite number, Characterization, Conjecture.

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25

Mahato, Prabhat, and Aayush Shah. "A Review of Prime Numbers, Squaring Prime Pattern, Different Types of Primes and Prime Factorization Analysis." International Journal for Research in Applied Science and Engineering Technology 11, no. 7 (July 31, 2023): 2036–43. http://dx.doi.org/10.22214/ijraset.2023.54904.

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Abstract: The study of prime numbers and their properties has always been an intriguing and fascinating topic for mathematicians. Primes can be considered the “basic building blocks,” the atoms, of the natural numbers. They play a significant role in number theory. Also, prime numbers, in this current world of computers and digitalization, have paramount significance for the computer programmers and scientists to tackle relevant real-life problems. Since long time, many studies and researches have been conducted regarding prime numbers pattern. In this paper, a squaring prime pattern is presented. Moreover, fifteen different types of primes with their Python code to generate them is included within. In cryptographic encryption system, prime numbers play a major role for security systems in which prime factorization is necessary. Therefore, prime factorization of composite numbers using Sieve of Eratosthenes algorithm on different platforms and time analysis based on that has been presented in the paper. Also, factorization analysis of primes by five different algorithms has been shown and comparison of prime factorization of composite numbers vs time taken graph has been plotted. Two major applications of primes are also covered.

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26

Gürefe, Nejla, and Gülfem Sarpkaya Aktaş. "The concept of prime number and the strategies used in explaining prime numbers." South African Journal of Education, no. 40(3) (August 31, 2020): 1–9. http://dx.doi.org/10.15700/saje.v40n3a1741.

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The teaching of mathematics does not only require the teacher to have knowledge about the subject, but the teacher also needs mathematical knowledge that is useful for the teaching and explaining thereof, as the teacher’s knowledge effects the students’ knowledge. A teacher should use appropriate mathematical explanation to be understood well by her/his students. In the study reported on here we investigated how prospective mathematics teachers defined the concept of prime number and which strategies they employed to explain the concept. The study was a descriptive survey within qualitative research. Forty-eight participants took part in the study and all completed the abstract algebra courses where they learned about the concept in question. The data collection tool was a form comprising 3 open-ended questions challenging what the concept of prime number was and how this concept could be explained to secondary/high school students. The data were analysed and the results show that the preservice teachers experienced great difficulty in defining the concept of prime number and that they used rules to explain prime numbers.

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27

Broughan, Kevin A., and Rory J. Casey. "Harmonic sets and the harmonic prime number theorem." Bulletin of the Australian Mathematical Society 71, no. 1 (February 2005): 127–37. http://dx.doi.org/10.1017/s0004972700038089.

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We restrict primes and prime powers to sets . Let . Then the error in θH(x) has, unconditionally, the expected order of magnitude . However, if then ψH (x) = x log 2 + O (log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the “harmonic prime number theorem”, πH (x)/π (x) → log 2.

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28

Matthews, C. R. "Matrix prime number theorems." Proceedings of the Japan Academy, Series A, Mathematical Sciences 65, no. 10 (1989): 336–39. http://dx.doi.org/10.3792/pjaa.65.336.

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29

Martin, Greg, and Nathan Ng. "Inclusive prime number races." Transactions of the American Mathematical Society 373, no. 5 (February 19, 2020): 3561–607. http://dx.doi.org/10.1090/tran/7996.

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30

Friedlander, John B., and Henryk Iwaniec. "Hyperbolic prime number theorem." Acta Mathematica 202, no. 1 (2009): 1–19. http://dx.doi.org/10.1007/s11511-009-0033-z.

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31

Welvaert, Marijke, Fernand Farioli, and Jonathan Grainger. "Graded Effects of Number of Inserted Letters in Superset Priming." Experimental Psychology 55, no. 1 (January 2008): 54–63. http://dx.doi.org/10.1027/1618-3169.55.1.54.

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Abstract. Three masked priming experiments investigated the effects of target word length and number of inserted letters on superset priming, where irrelevant letters are added to targets to form prime stimuli (e.g., tanble-table). Effects of one, two, three, and four-letter insertions were measured relative to an unrelated prime condition, the identity prime condition, and a condition where the order of letters of the superset primes was reversed. Superset primes facilitated performance compared with unrelated primes and reversed primes, and the overall pattern showed a small cost of letter insertion that was independent of target word length and that increased linearly as a function of the number of inserted letters. A meta-analysis incorporating data from the present study and two other studies investigating superset priming, showed an average estimated processing cost of 11 ms per letter insertion. Models of letter position coding are examined in the light of this result.

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32

Wójcik, J. "On a problem in algebraic number theory." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 2 (February 1996): 191–200. http://dx.doi.org/10.1017/s0305004100074090.

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Let K be an algebraic number field. If q is a prime ideal of the ring of integers of K and α is a number of K prime to q then Mq(α) denotes the multiplicative group generated by α modulo q. In the paper [5] there is the remark: ‘We do not know whether for all a, b, c ∈ ℚ with abc ≠ 0, |a| ≠ 1,|b| ≠ 1,|c| ≠ 1 there exist infinitely many primes q with Mq (a) = Mq (b) = Mq (c).’

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Reynvoet, Bert, Marc Brysbaert, and Wim Fias. "Semantic priming in number naming." Quarterly Journal of Experimental Psychology Section A 55, no. 4 (October 2002): 1127–39. http://dx.doi.org/10.1080/02724980244000116.

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The issue of semantic and non-semantic conversion routes for numerals is still debated in numerical cognition. We report two number-naming experiments in which the target numerals were preceded by another numeral (prime). The primes and targets could be presented either in arabic (digit) notation or in verbal (alphabetical) notation. The results reveal a semantically related distance effect: Latencies are fastest when the prime has the same value as the target and increase when the distance between prime and target increases. We argue that the present results are congruent with the idea that the numerals make access to an ordered semantic number line common to all notations, as the results are the same for within-notation priming (arabic-arabic or verbal- verbal) and between-notations priming (arabic-verbal or verbal-arabic). The present results also point to a rapid involvement of semantics in the naming of numerals, also when the numerals are words. As such, they are in line with recent claims of rapid semantic mediation in word naming.

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Banerjee, Kumarjit, Satyendra Nath Mandal, and Sanjoy Kumar Das. "A Comparative Study of Different Techniques for Prime Testing in Implementation of RSA." American Journal of Advanced Computing 1, no. 1 (January 1, 2020): 1–7. http://dx.doi.org/10.15864/ajac.1102.

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The RSA cryptosystem, invented by Ron Rivest, Adi Shamir and Len Adleman was first publicized in the August 1977 issue of Scientific American. The security level of this algorithm very much depends on two large prime numbers. The large primes have been taken by BigInteger in Java. An algorithm has been proposed to calculate the exact square root of the given number. Three methods have been used to check whether a given number is prime or not. In trial division approach, a number has to be divided from 2 to the half the square root of the number. The number will be not prime if it gives any factor in trial division. A prime number can be represented by 6n±1 but all numbers which are of the form 6n±1 may not be prime. A set of linear equations like 30k+1, 30k+7, 30k+11, 30k+13, 30k+17, 30k+19, 30k+23 and 30k+29 also have been used to produce pseudo primes. In this paper, an effort has been made to implement all three methods in implementation of RSA algorithm with large integers. A comparison has been made based on their time complexity and number of pseudo primes. It has been observed that the set of linear equations, have given better results compared to other methods.

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35

Malakhovsky, V. S. "About finding of prime numbers that follows after given prime number without using computer." Differential Geometry of Manifolds of Figures, no. 51 (2020): 81–85. http://dx.doi.org/10.5922/0321-4796-2020-51-10.

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It is shown how to define one or several prime numbers following after given prime number without using computer only by calculating several arithmetic progressions. Five examples of finding such prime numbers are given.

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Hu, Zhi Hua, and Qing Zhang. "The Strong Prime Numbers Generation Algorithm Based on Genetic Algorithm." Applied Mechanics and Materials 220-223 (November 2012): 2963–67. http://dx.doi.org/10.4028/www.scientific.net/amm.220-223.2963.

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Large prime number generation methods in need of a more complex modular exponentiation, leading to defects of slower computing speed, Based on this genetic algorithm, proposed a new strong prime number generated algorithm. The method according to the characteristics of Strong Primes, the algorithm is simple, easy to implement to meet the needs of the RSA algorithm security , giving the method of determining the large prime numbers. Design fitness function , crossover and mutation strategies which can be used in genetic algorithm. Finally design the algorithm of producing Strong prime numbers

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37

Hamiss, Kabenge. "A Simple Algorithm for Prime Factorization and Primality Testing." Journal of Mathematics 2022 (December 15, 2022): 1–10. http://dx.doi.org/10.1155/2022/7034529.

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We propose a new simple and faster algorithm to factor numbers based on the nature of the prime numbers contained in such composite numbers. It is well known that every composite number has a unique representation as a product of prime numbers. In this study, we focus mainly on composite numbers that contain a product of prime numbers that are greater than or equal to 5 which are of the form 6 k + 1 or 6 k + 5 . Therefore, we use the condition that every prime or composite P of primes greater than or equal to 5 satisfies P 2 ≡ 1 mod 24 . This algorithm is very fast especially when the difference in the prime components of a composite number (prime gap) is not so large. When the difference between the factors (prime gap) is not so large, it often requires just a single iteration to obtain the factors.

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Barghi, Mohammadreza. "A New Conjecture: Any Whole Number Greater than 3 has the Equal Distance from Two Prime Numbers." International Journal of Science and Research (IJSR) 12, no. 10 (October 5, 2023): 540–41. http://dx.doi.org/10.21275/sr231006065849.

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KANE, DANIEL M. "AN ASYMPTOTIC FOR THE NUMBER OF SOLUTIONS TO LINEAR EQUATIONS IN PRIME NUMBERS FROM SPECIFIED CHEBOTAREV CLASSES." International Journal of Number Theory 09, no. 04 (May 7, 2013): 1073–111. http://dx.doi.org/10.1142/s1793042113500139.

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We extend results relating to Vinogradov's three primes theorem to provide asymptotic estimates for the number of solutions to a given linear equation in three or more prime numbers under the additional constraint that each of the primes involved satisfies specialized Chebotarev conditions. In particular, we show that such solutions can be expected to exist unless a solution would violate some local constraint.

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Casinillo, Leomarich F. "Some New Notes on Mersenne Primes and Perfect Numbers." Indonesian Journal of Mathematics Education 3, no. 1 (April 30, 2020): 15. http://dx.doi.org/10.31002/ijome.v3i1.2282.

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<p>Mersenne primes are specific type of prime numbers that can be derived using the formula <img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" /> is a prime number. A perfect number is a positive integer of the form <img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&space;P(p)=2^{p-1}(2^{p}-1)" alt="" /> where <img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" /> is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" />. Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.</p>

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Perucca, Antonella, and Pietro Sgobba. "Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II." Uniform distribution theory 15, no. 1 (June 1, 2020): 75–92. http://dx.doi.org/10.2478/udt-2020-0004.

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AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

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42

Ghanouchi, Jamel. "A New Approach of the Concept of Prime Number." International Frontier Science Letters 2 (October 2014): 12–15. http://dx.doi.org/10.18052/www.scipress.com/ifsl.2.12.

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Gueye, Ibrahima. "Twin Primes and Sophie Germain’s Prime Numbers." Bulletin of Society for Mathematical Services and Standards 6 (June 2013): 1–3. http://dx.doi.org/10.18052/www.scipress.com/bsmass.6.1.

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For two millennia, the prime numbers have continued to fascinate mathematicians. Indeed, a conjecture which dates back to this period states that the number of twin primes is infinite. In 1949 Clement showed a theorem on twin primes. In a recent article, I prooved a corollary of Clement’s theorem [1]. In this paper, I will proove shortly the link between twin primes and Sophie Germain’s prime numbers.

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44

Banks, William D., and Igor E. Shparlinski. "On values taken by the largest prime factor of shifted primes." Journal of the Australian Mathematical Society 82, no. 1 (February 2007): 133–47. http://dx.doi.org/10.1017/s1446788700017511.

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AbstractLet P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n > 1. We show that, for every real number , there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c(η) = 1+ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P(q − a) ≍ qθ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P(q - a) for a > 0, and show that for infinitely many primes q, this map can be iterated at least (log logq)1+o(1) times before it terminates.

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45

Ghanouchi, Jamel. "The Concept of Prime Number and the Legendre Conjecture." International Frontier Science Letters 3 (January 2015): 16–18. http://dx.doi.org/10.18052/www.scipress.com/ifsl.3.16.

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46

Ascarelli, Paolo. "Detection and Rarefaction of the Twin Primes Numbers." European Journal of Mathematics and Statistics 4, no. 2 (April 11, 2023): 36–37. http://dx.doi.org/10.24018/ejmath.2023.4.2.216.

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In this manuscript are considered 3 types of numbers: a) integral numbers like for example (x)=10^10 b) prime numbers whose properties is to be only divisible by themselves c) twin numbers The number of twin primes contained under the number (x) is here derived by: 1) a mathematical function proposed by Gauss (1792-1796) based on a converging logarithmic sum, 2) Euclid’s theorems on prime numbers.

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47

Agdgomelashvili, Zurab. "Some interesting tasks from the classical number theory." Works of Georgian Technical University, no. 4(518) (December 15, 2020): 150–88. http://dx.doi.org/10.36073/1512-0996-2020-4-150-188.

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The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q is an odd prime number, and p N \{1}, p]1;q] [q  2; 2q] , then each of the different prime divisors of the number 1 1 pq A p    taken separately is greater than p; Theorem 4. Let’s q is an odd prime number, and p{q1; 2q1}, then from different prime divisors of the number 1 1 pq A p    taken separately, only one of them is less than p. A has at least two different simple divisors. Task 1. Solve the equation 1 2 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 2. Solve the equation 1 3 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 3. Solve the equation 1 1 z x y p y    where p{6; 7; 11; 13;} are the prime numbers, x, y  N and y is a prime number. There is a lema with which the problem class can be easily solved: Lemma ●. Let’s a, b, nN and (a,b) 1. Let’s prove that if an  0 (mod| ab|) , or bn  0 (mod| ab|) , then | ab|1. Let’s solve the equations ( – ) in natural x , y numbers: I. 2 z x y z z x y          ; VI. (x  y)xy  x y ; II. (x  y)z  (2x)z  yz ; VII. (x  y)xy  yx ; III. (x  y)z  (3x)z  yz ; VIII. (x  y) y  (x  y)x , (x  y) ; IV. ( y  x)x y  x y , (y  x) ; IX. (x  y)x y  xxy ; V. ( y  x)x y  yx , (y  x) ; X. (x  y)xy  (x  y)x , (y  x) . Theorem . If a, bN (a,b) 1, then each of the divisors (a2  ab  b2 ) will be similar. The concept of pseudofibonacci numbers is introduced and some of their properties are found.

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Lebowitz-Lockard, Noah. "Additively unique sets of prime numbers." International Journal of Number Theory 14, no. 10 (October 25, 2018): 2757–65. http://dx.doi.org/10.1142/s179304211850166x.

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Spiro proved that the identity function is the only multiplicative function with [Formula: see text] for some prime [Formula: see text] and [Formula: see text] for all prime [Formula: see text] and [Formula: see text]. We determine the sets [Formula: see text] of primes for which restricting our condition to [Formula: see text] for all [Formula: see text] still implies that [Formula: see text] is the identity function. We prove that [Formula: see text] satisfies these conditions if and only if [Formula: see text] contains every prime that is not the larger element of a twin prime pair and [Formula: see text] contains [Formula: see text] or [Formula: see text].

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49

Fiorilli, Daniel. "Highly biased prime number races." Algebra & Number Theory 8, no. 7 (October 21, 2014): 1733–67. http://dx.doi.org/10.2140/ant.2014.8.1733.

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50

Ashwell, Paul. "Prime concerns: painting number patterns." Journal of Mathematics and the Arts 14, no. 1-2 (April 2, 2020): 8–10. http://dx.doi.org/10.1080/17513472.2020.1734427.

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