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Journal articles on the topic 'Square of a number'

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Published: 5 June 2025

Last updated: 8 July 2025

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1

Coumbe, D. N. "MAGIC SQUARES OF PERFECT SQUARES AND PELL NUMBERS." JP Journal of Algebra, Number Theory and Applications 63, no. 6 (2024): 587–614. http://dx.doi.org/10.17654/0972555524032.

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Order three magic squares of distinct squared integers are studied. We show that such a magic square is not possible if the smallest entry is the square of a prime number, or unity. A method for generating all arithmetic progressions of three squared integers whose smallest term is the square of a prime or unity is presented via a set of linear transformation matrices involving the Pell numbers.

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2

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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3

Huxley, M. N., and O. Trifonov. "The square-full numbers in an interval." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 2 (1996): 201–8. http://dx.doi.org/10.1017/s0305004100074107.

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A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

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4

KÖKEN, Fikri, and Emre KANKAL. "Altered Numbers of Fibonacci Number Squared." Journal of New Theory, no. 45 (December 31, 2023): 73–82. http://dx.doi.org/10.53570/jnt.1368751.

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We investigate two types of altered Fibonacci numbers obtained by adding or subtracting a specific value $\{a\}$ from the square of the $n^{th}$ Fibonacci numbers $G^{(2)}_{F(n)}(a)$ and $H^{(2)}_{F(n)}(a)$. These numbers are significant as they are related to the consecutive products of the Fibonacci numbers. As a result, we establish consecutive sum-subtraction relations of altered Fibonacci numbers and their Binet-like formulas. Moreover, we explore greatest common divisor (GCD) sequences of r-successive terms of altered Fibonacci numbers represented by $\left\{G^{(2)}_{F(n), r}(a)\right\}$ and $\left\{H^{(2)}_{F(n), r}(a)\right\}$ such that $r\in\{1,2,3\}$ and $a\in\{1,4\}$. The sequences are based on the GCD properties of consecutive terms of the Fibonacci numbers and structured as periodic or Fibonacci sequences.

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5

Jakimczuk, Rafael. "Square-full numbers with an even number of prime factors." Notes on Number Theory and Discrete Mathematics 26, no. 1 (2020): 21–30. http://dx.doi.org/10.7546/nntdm.2020.26.1.21-30.

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6

Nurohmah, Siti, Dwiani Listya Kartika, Nuraini Muhassanah, and Dwi Ariani Finda Yuniarti. "Prediksi Jumlah Calon Siswa Baru Menggunakan Metode Double Exponential Smoothing Brown (Studi Kasus: SMK Ma’arif NU 1 Purwokerto Tahun Pelajaran 2024/2025)." Square : Journal of Mathematics and Mathematics Education 6, no. 1 (2024): 1–14. http://dx.doi.org/10.21580/square.2024.6.1.21263.

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The number of new students registering at SMK Ma'arif NU 1 Purwokerto from 2010 to 2023 has increased and decreased. This fluctuating pattern is a problem faced in determining strategies to attract new students to register. Large fluctuations in student numbers can cause instability in school revenues, which affects all operational aspects and will impact the school's reputation in the eyes of the community and prospective students. In this regard, it is necessary to take action, namely forecasting the number of new students registering for the 2024/2025 academic year. The aim of this research is to determine the prediction of the number of prospective new students at SMK Ma'arif NU 1 Purwokerto using theDouble Exponential Smoothing Brown in the 2024/2025 academic year. From the research results, it was obtained that the number of new students in the 2024/2025 academic year was based on overall data on students who registered with the best α parameter α = 0.2 with a value ofMAD = 6,02, MSE = 68,37, andMAP= 20.23%, namely 42 students. The forecast for the number of new students registering at SMK Ma'arif NU 1 Purwokerto based on overall data from the school has increased from the previous year. Value resultsMAPshows that the forecasting that has been carried out falls into the category of adequate forecasting method capabilities so that it can be used to predict the number of new students enrolling at SMK Ma'arif NU 1 Purwoketo and the predicted value for the following year can be known.Keywords: predictions, forecasting, new students, Double Exponential Smoothing Brown

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7

Shang Gao, Zaiyue Zhang, and Cungen Cao. "Square Operation of Triangular Fuzzy Number." INTERNATIONAL JOURNAL ON Advances in Information Sciences and Service Sciences 4, no. 14 (2012): 16–24. http://dx.doi.org/10.4156/aiss.vol4.issue14.3.

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8

Klamkin, M. S., T. Lewis, and A. Liu. "The Kissing Number of the Square." Mathematics Magazine 68, no. 2 (1995): 128. http://dx.doi.org/10.2307/2691191.

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9

Klamkin, M. S., T. Lewis, and A. Liu. "The Kissing Number of the Square." Mathematics Magazine 68, no. 2 (1995): 128–33. http://dx.doi.org/10.1080/0025570x.1995.11996296.

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10

Dongmei, Ren. "On the square-free number sequence." Scientia Magna Vol. 1, No. 2 (2005): 46–48. https://doi.org/10.5281/zenodo.9297.

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The main purpose of this paper is to study the number of the square-free number sequence, and give two interesting asymptotic formulas for it. At last, give another asymptotic formula and a corollary.

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11

Ciliberto, Ciro, and Rick Miranda. "Nagata’s conjecture for a square or nearly-square number of points." Ricerche di Matematica 55, no. 1 (2006): 71–78. http://dx.doi.org/10.1007/s11587-006-0005-y.

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12

Mirji, Praveen. "Modal Analysis of a Square Plate with Reinforcement with Number of Stiffeners." International Journal of Trend in Scientific Research and Development Volume-3, Issue-3 (2019): 1157–62. http://dx.doi.org/10.31142/ijtsrd23207.

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13

Neeradha., C. K.* Dr. T.S Sivakumar Dr.V. Madhukar Mallayya. "MATHEMATICAL PROPERTIES OF 9×9 STRONGLY MAGIC SQUARES." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 4 (2017): 230–36. https://doi.org/10.5281/zenodo.546315.

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A magic square is a square array of numbers where the rows, columns, diagonals and co-diagonals add up to the same number. The paper discusses about a well-known class of magic squares; the strongly magic square. The strongly magic square is a magic square with a stronger property that the sum of the entries of the sub-squares taken without any gaps between the rows or columns is also the magic constant. In this paper a generic definition for Strongly Magic Squares is given. This paper focuses on discovering the mathematical properties of 9×9 Strongly Magic Squares

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14

Chan, Heng Huat, Shaun Cooper, and Wen-Chin Liaw. "An odd square as a sum of an odd number of odd squares." Acta Arithmetica 132, no. 4 (2008): 359–71. http://dx.doi.org/10.4064/aa132-4-5.

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15

Al-Zhour, Zeyad. "The Total Number of Squares and Rectangles are There in Rectangle ( Square ) Boards." International Journal of Open Problems in Computer Science and Mathematics 5, no. 3 (2012): 152–59. http://dx.doi.org/10.12816/0006132.

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16

Sabani, Ade Mahmud Eka. "Using the Factor Tree Strategy to Overcome Learning Difficulties for Junior High School Students in Helping to Solve Square Roots." Journal of Medives : Journal of Mathematics Education IKIP Veteran Semarang 7, no. 1 (2023): 1. http://dx.doi.org/10.31331/medivesveteran.v7i1.1850.

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The problem in the research is the difficulty of students in solving problems of the square root of integers (perfect squares) and cube roots of integers (cubic numbers) using an algorithm strategy better known as porogapit. This problem was obtained when giving tutoring to several junior high school students in Purworejo. The algorithm or porogapit strategy was taught when I was in elementary school. Still, as the learning model developed, the strategy could have been more efficient and took a long time. Only a few students remembered the strategy. Today's students even use a trial-and-error strategy: try the same multiplication number one by one or raise the numbers sequentially until the number in question. The trial and error strategy solved only the square root of integers (perfect square). In contrast, when there was a problem with the square root of imperfect integers (irrational squares), they still had difficulty solving it. The results showed that it was easier for students to master the factor tree strategy in solving square root problems than the algorithmic and trial-and-error strategies. Furthermore, the factor tree strategy can assist students in solving the square root problem, both the square root of perfect integers and the square root of imperfect numbers (irrational squares). Keywords: Students Learning Difficulties, Square Root Solutions.

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17

Kim, Aeran. "Square Fibonacci Numbers and Square Lucas Numbers." Asian Research Journal of Mathematics 3, no. 3 (2017): 1–8. http://dx.doi.org/10.9734/arjom/2017/32312.

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18

Changa, M. E. "On the number of primes yielding square-free sums with given numbers." Russian Mathematical Surveys 58, no. 3 (2003): 613–14. http://dx.doi.org/10.1070/rm2003v058n03abeh000637.

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19

Anindita, Henindya Permatasari, and Dian Pratama Jovian. "HIGH ORDER ANTI EVEN LEAST SQUARE FOR APPROXIMATING PRIME NUMBERS BELOW 1000." International Journal of Mathematics and Computer Research 12, no. 06 (2024): 4315–19. https://doi.org/10.5281/zenodo.12180292.

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The study of prime numbers, pivotal in mathematics for centuries, holds significant importance in number theory and diverse applications like cryptography and computer science. This article introduces a novel approach, "High Order Anti Even Least Square," for approximating prime numbers below 1000. Integrating Least Square with specialized techniques tailored to complex prime number distributions, this method aims to enhance accuracy compared to traditional approaches. The research methodology involves polynomial approximation using both traditional and anti-even least squares methods, error calculation, visualization, and analysis. Results indicate that while traditional least squares generally performs better, the anti-even least squares method shows promise, particularly at higher polynomial degrees. This study contributes to advancing number theory and its practical applications by presenting a novel method for prime number approximation.

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20

Barbeau, E. J. "Numbers Differing from Consecutive Squares by Squares." Canadian Mathematical Bulletin 28, no. 3 (1985): 337–42. http://dx.doi.org/10.4153/cmb-1985-040-9.

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AbstractIt is shown that there are infinitely many natural numbers which differ from the next four greater perfect squares by a perfect square. This follows from the determination of certain families of solutions to the diophantine equation 2(b2 + 1) = a2 + c2. However, it is essentially known that any natural number with this property cannot be 1 less than a perfect square. The question whether there exists a number differing from the next five greater squares by squares is open.

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21

Gopalan, M. A., and V. Geetha. "M-Gonal number ±3 = A Perfect square." International Journal of Mathematics Trends and Technology 17, no. 1 (2015): 32–35. http://dx.doi.org/10.14445/22315373/ijmtt-v17p506.

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22

Győri, Ervin, Gyula Y. Katona, and László F. Papp. "Optimal Pebbling Number of the Square Grid." Graphs and Combinatorics 36, no. 3 (2020): 803–29. http://dx.doi.org/10.1007/s00373-020-02154-z.

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23

Chegini, A. G., Morteza Hasanvand, E. S. Mahmoodian, and Farokhlagha Moazami. "The square chromatic number of the torus." Discrete Mathematics 339, no. 2 (2016): 447–56. http://dx.doi.org/10.1016/j.disc.2015.09.003.

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24

Müller, Wolfgang R., Klaus Szymanski, Jan V. Knop, and Nenad Trinajstić. "On the number of square-cell configurations." Theoretica Chimica Acta 86, no. 3 (1993): 269–78. http://dx.doi.org/10.1007/bf01130823.

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25

Gouvea, F. Q. "The Square-Free Sieve over Number Fields." Journal of Number Theory 43, no. 1 (1993): 109–22. http://dx.doi.org/10.1006/jnth.1993.1012.

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26

Shabanifar, Samad, and Alper Cihan Konyalioglu. "Mathematics teachers’ approaches to students’ possible mistakes in exponential and square root number." International Journal of Academic Research 5, no. 6 (2013): 213–19. http://dx.doi.org/10.7813/2075-4124.2013/5-6/b.34.

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27

Hooley, Christopher. "On the representation of a number as the sum of a prime and two squares of square-free numbers." Acta Arithmetica 182, no. 3 (2018): 201–29. http://dx.doi.org/10.4064/aa8514-9-2016.

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28

Dambrosia, Barbara K. "Square-free rings." Communications in Algebra 27, no. 5 (1999): 2045–71. http://dx.doi.org/10.1080/00927879908826549.

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29

Hulse, Thomas A., Chan Ieong Kuan, Eren Mehmet Kıral, and Li-Mei Lim. "Counting square discriminants." Journal of Number Theory 162 (May 2016): 255–74. http://dx.doi.org/10.1016/j.jnt.2015.10.015.

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30

Diana, Arista Fitri, Muhammad Ibnu Hajar, Zakaria Bani Ikhtiyar, and Lathifatul Aulia. "Analisis Kestabilan Lokal Model Transmisi Demam Berdarah Dengue." Square : Journal of Mathematics and Mathematics Education 6, no. 1 (2024): 41–54. http://dx.doi.org/10.21580/square.2024.6.1.21018.

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Dengue fever transmission in Indonesia has an advanced amount. In this article, dynamic model of interaction between human and Aedes aegypti mosquitos is learned. The SEIRRD (Susceptible, Exposed, Infected, Recovered, Deceased) model is used in this article. The prurpose in this model is to describe the stability of dengue transmission, so that we can analyze the developed of epidemic model in mathemtic field. Using NGM method to analyze basic reproduction number and applying Routh-Hurwitz criteria method to show the local stability of model. Then, two equilibrium points, called endemic and non-endemic equilibrium points, are obtained. The result of basic reproduction number is described the stability analysis. If basic reproduction number less then one, the endemic equilibrium point is locally asymptotically stable and otherwise. Local stability analysis at the equilibrium point is determined through parameter analysis. Furthermore, numerical simulations are carried out by fitting the data to obtaine the result of the parameters. The results of numerical simulations explaine the spread of dengue transmission Keywords: Dynamic Model, Epidemic Model, Equilibrium Point, Local Stability, Routh Hurwitz

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31

Hachez, Yvan, and Hugo J. Woerdeman. "Approximating sums of squares with a single square." Linear Algebra and its Applications 399 (April 2005): 187–201. http://dx.doi.org/10.1016/j.laa.2004.10.005.

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32

Ewbank, William A. "Number Tablets for Primary Grades." Arithmetic Teacher 36, no. 7 (1989): 20–23. http://dx.doi.org/10.5951/at.36.7.0020.

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Number tablets are simple and versatile teaching devices. These small square cards numbered 1–50 (or 1–20 for beginners) can be turned facedown, shuffled, and moved around in game and puzzle situations. The number tablets (referred to as NT hereafter) can be any suitable size, but I find that 1-inch (2.5-cm) squares are convenient. Over 600 NTs of this size can be cut from one standard sheet of posterboard, and pupils can write the numerals on the blank cards (fig. 1).

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33

Powers, Michael R., and Martin Shubik. "A "square-root rule" for reinsurance." Revista Contabilidade & Finanças 17, spe2 (2006): 101–7. http://dx.doi.org/10.1590/s1519-70772006000500008.

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In previous work, the authors derived a mathematical expression for the optimal (or "saturation") number of reinsurers for a given number of primary insurers (see Powers and Shubik, 2001). In the current article, we show analytically that, for large numbers of primary insurers, this mathematical expression provides a "square-root rule"; i.e., the optimal number of reinsurers in a market is given asymptotically by the square root of the total number of primary insurers. We note further that an analogous "fourth-root rule" applies to markets for retrocession (the reinsurance of reinsurance).

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34

Oboudi, Mohammad Reza. "A note on constructing and enumerating of magic squares." Boletim da Sociedade Paranaense de Matemática 40 (February 2, 2022): 1–4. http://dx.doi.org/10.5269/bspm.46836.

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Let $n\geq1$ be an integer. A magic square of order $n$ is a square table $n\times n$, say $A$, filled with distinct positive numbers $1,2,\ldots,n^2$ such that all cells of $A$ are distinct and the sum of the numbers in each row, column and diagonal is equal.Let $M(n,s)$ be the set of all $n\times n$ $(0,1)$-matrices, say $T$, such that the number of $1$ in every row and every column of $T$ is $s$.In this paper for every positive integer $k$ we find a new way for constructing magic squares of order $4k$. We show that the number of magic squares of order $4k$ is at least $|M(2k,k)|$. In particular we show that the number of magic squares of order $4k$ is at least $\frac{{2k \choose k}^2}{2}$.

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35

Steva, Stojilković. "Squaring the circle and number π". Journal of Progressive Research in Mathematics 15, № 3 (2019): 2695–99. https://doi.org/10.5281/zenodo.3974039.

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This paper presents an attempt to finally solve the problem of constructing the squares of the same surface of the given circle, in a simple way: by determining the exact value of the number π. The value of this constant thus far implies the following: in the Universe there is no square of the same surface as the given circle!?It's pointless. The number π is not irrational, so it is not transcendental number either.

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36

Jung, Sang Mok, Ji-Hye Seo, and Poo Gyeon Park. "Normalized Least-Mean-Square Algorithm with a Pseudo-Fractional Number of Orthogonal Correction Factors." Journal of Advances in Computer Networks 3, no. 2 (2015): 167–72. http://dx.doi.org/10.7763/jacn.2015.v3.161.

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37

Jouis, Christophe, and Mercedes Orús-Lacort. "Value of the golden ratio (number Phi) knowing the side length of a square." Selecciones Matemáticas 8, no. 02 (2021): 404–10. http://dx.doi.org/10.17268/sel.mat.2021.02.16.

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38

Li, Jiancheng, Yali Li, Yunxiao Dong, Yuede Yang, Jinlong Xiao, and Yongzhen Huang. "400 Gb/s physical random number generation based on deformed square self-chaotic lasers." Chinese Optics Letters 21, no. 6 (2023): 061901. http://dx.doi.org/10.3788/col202321.061901.

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39

Chia, G. L., W. Hemakul, and S. Singhun. "Graphs with cyclomatic number three having panconnected square." Discrete Mathematics, Algorithms and Applications 09, no. 05 (2017): 1750067. http://dx.doi.org/10.1142/s1793830917500677.

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The square of a graph [Formula: see text] is the graph obtained from [Formula: see text] by adding edges joining those pairs of vertices whose distance from each other in [Formula: see text] is two. If [Formula: see text] is connected, then the cyclomatic number of [Formula: see text] is defined as [Formula: see text]. Graphs with cyclomatic number not more than [Formula: see text] whose square are panconnected have been characterized, among other things, in [G. L. Chia, S. H. Ong and L. Y. Tan, On graphs whose square have strong Hamiltonian properties, Discrete Math. 309 (2009) 4608–4613, G. L. Chia, W. Hemakul and S. Singhun, Graphs with cyclomatic number two having panconnected square, Discrete Math. 311 (2011) 850–855]. Here, we show that if [Formula: see text] has cyclomatic number [Formula: see text] and [Formula: see text] is panconnected, then [Formula: see text] is one of the eight families of graphs, [Formula: see text], defined in the paper. Further, we obtain necessary and sufficient conditions for three larger families of graphs (which contains [Formula: see text] as special cases) whose square are panconnected.

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40

Tho, Nguyen Xuan. "Two theorems on square numbers." Notes on Number Theory and Discrete Mathematics 28, no. 1 (2022): 75–80. http://dx.doi.org/10.7546/nntdm.2022.28.1.75-80.

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We show that if a is a positive integer such that for each positive integer n, a+n^2 can be expressed x^2+y^2, where x,y\in \mathbb{Z}, then a is a square number. A similar theorem also holds if a+n^2 and x^2+y^2 are replaced by a+2n^2 and x^2+2y^2, respectively.

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41

Gohberg, I., and T. Shalom. "On inversion of square matrices partitioned into non-square blocks." Integral Equations and Operator Theory 12, no. 4 (1989): 539–66. http://dx.doi.org/10.1007/bf01199458.

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42

ŞAHİN, Adem. "Square numbers, square pyramidal numbers, and generalized Fibonacci polynomials." Journal of New Results in Science 11, no. 1 (2022): 91–99. http://dx.doi.org/10.54187/jnrs.1105346.

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43

LI, YAN, and LIANRONG MA. "DOUBLE COVERINGS AND UNIT SQUARE PROBLEMS FOR CYCLOTOMIC FIELDS." International Journal of Number Theory 07, no. 07 (2011): 1935–44. http://dx.doi.org/10.1142/s1793042111004836.

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In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for [Formula: see text], where K = ℚ(ζn), G = Gal (K/ℚ), 𝔽2 = ℤ/2ℤ and UK is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζn) such that [Formula: see text]. Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, J. Number Theory128 (2008) 2687–2694]. In particular, we prove that the unit square problem does not hold for ℚ(ζn) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields ℚ(ζn)v with v | p but not a square in ℚ(ζn), if n has more than three distinct prime factors.

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44

Budiman, Arif, Efori Bulolo, and Imam Saputra. "Middle Square Method Analysis of Number Pseudorandom Process." IJICS (International Journal of Informatics and Computer Science) 4, no. 2 (2020): 35. http://dx.doi.org/10.30865/ijics.v4i2.1386.

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Random numbers can be generated from a calculation of mathematical formulas. Such random numbers are often referred to as pseudo random numbers, random numbers are used for various algorithms, especially cryptographic algorithms such as AES, RSA, IDEA, GOST that require the use of Middle-Square Method random numbers which is very useful for adding research references to algorithms concerning random number generator and better understand how random numbers are generated using the Middle-Square Method algorithm. Both data collection and report making as for the objectives achieved in the form of understanding random numbers and knowing the algorithm process, compile a program designed to be used as an alternative to random numbers for various purposes, especially in cryptographic algorithms.

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45

Zebib, A., G. M. Homsy, and E. Meiburg. "High Marangoni number convection in a square cavity." Physics of Fluids 28, no. 12 (1985): 3467. http://dx.doi.org/10.1063/1.865300.

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46

Jakhar, Anuj, Sudesh K. Khanduja, and Neeraj Sangwan. "Discriminants of pure square-free degree number fields." Acta Arithmetica 181, no. 3 (2017): 287–96. http://dx.doi.org/10.4064/aa170508-4-11.

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Billenness, D. A., N. Djilali, and E. Zeidan. "LOW REYNOLDS NUMBER FLOW OVER A SQUARE RIB." Transactions of the Canadian Society for Mechanical Engineering 21, no. 4 (1997): 371–87. http://dx.doi.org/10.1139/tcsme-1997-0018.

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Laminar flow over a square rib placed in a fully developed channel flow is investigated over the Reynolds number range 80-350. The effect of Reynolds number on the flow and the variation of the primary reattachment length with Reynolds number are investigated using flow visualization and laser-Doppler velocimetry. The primary recirculation region length is found to increase in a slightly non-linear fashion with Reynolds number up to Reh = 250, at which point shear layer instabilities first appear downstream of the rib. Increasing the Reynolds number further, first results in continuing growth of the separation bubble, and then for Reh ≳ 300, in the appearance of three dimensional vortices and gradual shortening of the bubble. The measurements are complemented by two- and three-dimensional numerical simulations using a finite volume method with a high-order descretization scheme. These simulations yield excellent agreement with the measured reattachment lengths and velocity profiles over the steady laminar flow régime.

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48

Bai, Honglei, and Md Mahbub Alam. "Dependence of square cylinder wake on Reynolds number." Physics of Fluids 30, no. 1 (2018): 015102. http://dx.doi.org/10.1063/1.4996945.

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McKay, Brendan D., Jeanette C. McLeod, and Ian M. Wanless. "The number of transversals in a Latin square." Designs, Codes and Cryptography 40, no. 3 (2006): 269–84. http://dx.doi.org/10.1007/s10623-006-0012-8.

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50

Besharati, Nazli, Luis Goddyn, E. S. Mahmoodian, and M. Mortezaeefar. "On the chromatic number of Latin square graphs." Discrete Mathematics 339, no. 11 (2016): 2613–19. http://dx.doi.org/10.1016/j.disc.2016.04.025.

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