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Journal articles on the topic 'Sum of two squares'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

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1

Roy, S. C. Dutta. "Approximating the Square Root of the Sum of Two Squares." IETE Journal of Education 32, no. 2 (1991): 11–13. http://dx.doi.org/10.1080/09747338.1991.11436322.

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2

Gutierrez, Jaime, Álvar Ibeas, and Antoine Joux. "Recovering a sum of two squares decomposition." Journal of Symbolic Computation 64 (August 2014): 16–21. http://dx.doi.org/10.1016/j.jsc.2013.12.003.

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3

Hourong, Qin. "The sum of two squares in a quadratic field." Communications in Algebra 25, no. 1 (1997): 177–84. http://dx.doi.org/10.1080/00927879708825844.

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4

Dresden, Greg, Kylie Hess, Saimon Islam, et al. "When is an+ 1 the sum of two squares ?" Involve, a Journal of Mathematics 12, no. 4 (2019): 585–605. http://dx.doi.org/10.2140/involve.2019.12.585.

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5

Troupe, Lee. "Divisor sums representable as the sum of two squares." Proceedings of the American Mathematical Society 148, no. 10 (2020): 4189–202. http://dx.doi.org/10.1090/proc/15104.

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6

Crocker, Roger Clement. "On the sum of two squares and two powers of k." Colloquium Mathematicum 112, no. 2 (2008): 235–67. http://dx.doi.org/10.4064/cm112-2-3.

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7

van Ginkel, Joost R., and Pieter M. Kroonenberg. "Much ado about nothing: Multiple imputation to balance unbalanced designs for two-way analysis of variance." Methodology 16, no. 4 (2020): 335–53. http://dx.doi.org/10.5964/meth.4327.

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In earlier literature, multiple imputation was proposed to create balance in unbalanced designs, as an alternative to Type III sum of squares in two-way ANOVA. In the current simulation study we studied four pooled statistics for multiple imputation, namely D₀, D₁, D₂, and D₃ in unbalanced data, and compared these statistics with Type III sum of squares. Statistics D₀ and D₂ generally performed best regarding Type-I error rates, and had power rates closest to that of Type III sum of squares. However, none of the statistics produced power rates higher than Type III sum of squares. The results l

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8

van Ginkel, Joost R., and Pieter M. Kroonenberg. "Multiple imputation to balance unbalanced designs for two-way analysis of variance." Methodology 17, no. 1 (2021): 39–57. http://dx.doi.org/10.5964/meth.6085.

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A balanced ANOVA design provides an unambiguous interpretation of the F-tests, and has more power than an unbalanced design. In earlier literature, multiple imputation was proposed to create balance in unbalanced designs, as an alternative to Type-III sum of squares. In the current simulation study we studied four pooled statistics for multiple imputation, namely D₀, D₁, D₂, and D₃ in unbalanced data, and compared them with Type-III sum of squares. Statistics D₁ and D₂ generally performed best regarding Type-I error rates, and had power rates closest to that of Type-III sum of squares. Additio

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9

David J. Platt and Timothy S. Trudgian. "On the Sum of Two Squares and At Most Two Powers of 2." American Mathematical Monthly 124, no. 8 (2017): 737. http://dx.doi.org/10.4169/amer.math.monthly.124.8.737.

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10

Gandhi, K. Raja Rama. "Primes of the form x²+ ny²." Bulletin of Society for Mathematical Services and Standards 3 (September 2012): 67–72. http://dx.doi.org/10.18052/www.scipress.com/bsmass.3.67.

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We know that, Fermat Showed a prime can be expressed as a sum of two squares if and only if it is a multiple of four plus one and its decomposition is unique. This paper will discuss the similar writings of primes as a sum of squares and multiple of another square.

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11

McLean, K. Robin. "85.55 What Proportion of Integers Are the Sum of Two Squares?" Mathematical Gazette 85, no. 504 (2001): 470. http://dx.doi.org/10.2307/3621757.

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12

Plaksin, V. A. "THE DISTRIBUTION OF NUMBERS REPRESENTABLE AS A SUM OF TWO SQUARES." Mathematics of the USSR-Izvestiya 31, no. 1 (1988): 171–91. http://dx.doi.org/10.1070/im1988v031n01abeh001054.

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13

Banks†, William D., Florian Luca‡, Filip Saidak§, and Igor E. Shparlinski¶. "Values of arithmetical functions equal to a sum of two squares." Quarterly Journal of Mathematics 56, no. 2 (2005): 123–39. http://dx.doi.org/10.1093/qmath/hah039.

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14

LIU, ZHIXIN, and GUANGSHI LÜ. "DENSITY OF TWO SQUARES OF PRIMES AND POWERS OF 2." International Journal of Number Theory 07, no. 05 (2011): 1317–29. http://dx.doi.org/10.1142/s1793042111004605.

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As the generalization of the problem of Romanoff, we establish that a positive proportion of integers can be written as the sum of two squares of primes and two powers of 2. We also prove that every large even integer can be written as the sum of two primes and 12 powers of 2.

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15

Kalmynin, Alexander. "INTERVALS BETWEEN CONSECUTIVE NUMBERS WHICH ARE SUMS OF TWO SQUARES." Mathematika 65, no. 4 (2019): 1018–32. http://dx.doi.org/10.1112/s0025579319000299.

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In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider a certain sum of Bessel functions and prove the upper bound for its mean value. This bound provides estimates for the $\unicode[STIX]{x1D6FE}$th moments of gaps for all $\unicode[STIX]{x1D6FE}\leqslant 2$.

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16

Roberts, Tim S. "The discovery of two new magic knight’s tours." Mathematical Gazette 89, no. 514 (2005): 22–27. http://dx.doi.org/10.1017/s0025557200176600.

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A magic square is one in which all rows and columns, and the two main diagonals, sum to the same total. A knight’s tour is a tour of the board in which, using knight’s moves, all squares are visited exactly once. When the squares visited are numbered from 1 to 64, if the square is magic (but without including the two main diagonals), this is termed a magic knight’s tour. This paper describes two magic knight’s tours on an 8 by 8 board found in early 2003, the first new tours to be discovered since 1988, and the first irregular tours to be discovered since 1936.

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17

Ibrahim, Amal Abdulrahman, and Shatha A. Salman. "Some properties of magSome Properties of Magic Squares of Distinct Squares and Cubesic squares." Al-Mustansiriyah Journal of Science 30, no. 3 (2019): 60. http://dx.doi.org/10.23851/mjs.v30i3.664.

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Magic squares is n×n matrix with positive integer entries as well as the sum of rows, columnsand mains diagonal have the same magic constant, one of the most oldest magic square wasdiscovered in china. In this paper the history of magic square is displayed and some definitionof its kind is given the prove of two theorems about properties of magic square is introduced.

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18

Fine, Benjamin. "Cyclotomic equations and square properties in rings." International Journal of Mathematics and Mathematical Sciences 9, no. 1 (1986): 89–95. http://dx.doi.org/10.1155/s016117128600011x.

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IfRis a ring, the structure of the projective special linear groupPSL2(R)is used to investigate the existence of sum of square properties holding inR. Rings which satisfy Fermat's two-square theorem are called sum of squares rings and have been studied previously. The present study considers a related property called square property one. It is shown that this holds in an infinite class of rings which includes the integers, polynomial rings over many fields andZpnwherePis a prime such that−3is not a squaremodp. Finally, it is shown that the class of sum of squares rings and the class satisfying

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19

Bleher, Pavel, and Freeman Dyson. "Mean square value of exponential sums related to representation of integers as sum of two squares." Acta Arithmetica 68, no. 1 (1994): 71–84. http://dx.doi.org/10.4064/aa-68-1-71-84.

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20

Blomer, V., J. Brüdern, and R. Dietmann. "Sums of smooth squares." Compositio Mathematica 145, no. 6 (2009): 1401–41. http://dx.doi.org/10.1112/s0010437x09004254.

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AbstractLet R(n,θ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding nθ/2. When θ>e−1/3 a lower bound for R(n,θ) of the expected order of magnitude is established, and when θ>365/592, it is shown that R(n,θ)>0 holds for large n. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed θ>0. This last result is the key to bound

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21

Balog, Antal, and Trevor D. Wooley. "Sums of Two Squares in Short Intervals." Canadian Journal of Mathematics 52, no. 4 (2000): 673–94. http://dx.doi.org/10.4153/cjm-2000-029-6.

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AbstractLet denote the set of integers representable as a sum of two squares. Since can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that hasmany properties in common with the set of prime numbers. In this paper we exhibit “unexpected irregularities” in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of than exp

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22

Languasco, Alessandro, and Alessandro Zaccagnini. "Sum of one prime and two squares of primes in short intervals." Journal of Number Theory 159 (February 2016): 45–58. http://dx.doi.org/10.1016/j.jnt.2015.07.010.

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23

CHOI, S. K. K., A. V. KUMCHEV, and R. OSBURN. "ON SUMS OF THREE SQUARES." International Journal of Number Theory 01, no. 02 (2005): 161–73. http://dx.doi.org/10.1142/s1793042105000054.

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Let r3(n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two alternative proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of r3(n).

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24

Dolan, Stan. "Thoughts on a conjecture of Erdős." Mathematical Gazette 101, no. 552 (2017): 449–57. http://dx.doi.org/10.1017/mag.2017.126.

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If two squares with no interior point in common are drawn inside a unit square then prove that the sum of their side-lengths is at most 1.This problem was posed in the 1930s by Paul Erdős [1]. It is the simplest case of a still unsolved conjecture.If k2 + 1 squares with no interior point in common are drawn inside a unit square then the maximum possible sum of their side-lengths is k [2].We shall use the notation S(n) to denote the maximum possible sum of the side-lengths for n squares drawn with no interior point in common inside a unit square. The main aim of this article will be to develop

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25

Liu, Huafeng. "Two results on powers of two in Waring–Goldbach type problems." International Journal of Number Theory 12, no. 07 (2016): 1813–25. http://dx.doi.org/10.1142/s1793042116501128.

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In this paper, it is proved that for [Formula: see text], every sufficient large odd integer is a sum of one prime, two squares of primes and [Formula: see text] powers of two. Furthermore, for [Formula: see text], every pair of large odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two squares of primes and [Formula: see text] powers of two. These improve the previous results [Formula: see text] and [Formula: see text].

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26

LIU, YUHUI. "ON SUMS OF TWO PRIME SQUARES, FOUR PRIME CUBES AND POWERS OF TWO." Bulletin of the Australian Mathematical Society 102, no. 2 (2020): 207–16. http://dx.doi.org/10.1017/s0004972719001382.

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We prove that every sufficiently large even integer can be represented as the sum of two squares of primes, four cubes of primes and 28 powers of two. This improves the result obtained by Liu and Lü [‘Two results on powers of 2 in Waring–Goldbach problem’, J. Number Theory 131(4) (2011), 716–736].

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27

Overmars, Anthony, and Sitalakshmi Venkatraman. "A Fast Factorisation of Semi-Primes Using Sum of Squares." Mathematical and Computational Applications 24, no. 2 (2019): 62. http://dx.doi.org/10.3390/mca24020062.

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For several centuries, prime factorisation of large numbers has drawn much attention due its practical applications and the associated challenges. In computing applications, encryption algorithms such as the Rivest–Shamir–Adleman (RSA) cryptosystems are widely used for information security, where the keys (public and private) of the encryption code are represented using large prime factors. Since prime factorisation of large numbers is extremely hard, RSA cryptosystems take advantage of this property to ensure information security. A semi-prime being, a product of two prime numbers, has wide a

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28

YIU, PAUL Y. H. "ON THE PRODUCT OF TWO SUMS OF 16 SQUARES AS A SUM OF SQUARES OF INTEGRAL BILINEAR FORMS." Quarterly Journal of Mathematics 41, no. 4 (1990): 463–500. http://dx.doi.org/10.1093/qmath/41.4.463.

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29

Zhai, Shuai. "Average behavior of Fourier coefficients of cusp forms over sum of two squares." Journal of Number Theory 133, no. 11 (2013): 3862–76. http://dx.doi.org/10.1016/j.jnt.2013.05.013.

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30

Wei, Dasheng. "On the sum of two integral squares in quadratic fields Q(\sqrt±p)." Acta Arithmetica 147, no. 3 (2011): 253–60. http://dx.doi.org/10.4064/aa147-3-5.

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31

Chan, Heng Huat, and Pee Choon Toh. "On a symmetric identity of Ramanujan involving the sum of two squares function." Journal of Mathematical Analysis and Applications 473, no. 2 (2019): 1234–43. http://dx.doi.org/10.1016/j.jmaa.2019.01.018.

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32

Daniel, Stephan. "On the Greatest Prime Divisor of the Sum of Two Squares of Primes." Journal of the London Mathematical Society 60, no. 3 (1999): 646–58. http://dx.doi.org/10.1112/s0024610799008042.

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33

Lü, Guangshi. "On sum of one prime, two squares of primes and powers of 2." Monatshefte für Mathematik 187, no. 1 (2017): 113–23. http://dx.doi.org/10.1007/s00605-017-1104-4.

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34

Plaksin, V. A. "Pairs of integers which can be written as the sum of two squares." Mathematical Notes of the Academy of Sciences of the USSR 40, no. 2 (1986): 579–85. http://dx.doi.org/10.1007/bf01159111.

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35

Kearsley, S. K. "On the orthogonal transformation used for structural comparisons." Acta Crystallographica Section A Foundations of Crystallography 45, no. 2 (1989): 208–10. http://dx.doi.org/10.1107/s0108767388010128.

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Rotation matrices that minimize or maximize the sum of the squared distances between corresponding atoms for two structures are found using a constrained least-squares procedure solved analytically as an eigenvalue problem in quaternion parameters.

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36

Choudhry, Ajai. "Sextuples of integers whose sums in pairs are squares." International Journal of Number Theory 11, no. 02 (2015): 543–55. http://dx.doi.org/10.1142/s1793042115500281.

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This paper is concerned with the diophantine problem of finding six integers such that the sum of any two of them is a perfect square. Till now, only one numerical example of such a sextuple has been published. In this paper, we obtain infinitely many examples of sextuples of integers such that the sum of any two of them is a perfect square. These examples include sextuples which have three or four or five distinct integers as well as sextuples in which all the integers are distinct.

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37

Kim, Daehak. "Derivation of error sum of squares of two stage nested designs and its application." Journal of the Korean Data and Information Science Society 24, no. 6 (2013): 1439–48. http://dx.doi.org/10.7465/jkdi.2013.24.6.1439.

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38

Plaksin, V. A. "THE NUMBER OF INTEGERS REPRESENTABLE AS A SUM OF TWO SQUARES ON SMALL INTERVALS." Mathematics of the USSR-Izvestiya 28, no. 1 (1987): 67–78. http://dx.doi.org/10.1070/im1987v028n01abeh000867.

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39

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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40

Ladoucette, Sophie, and Jef Teugels. "Limit distributions for the ratio of the random sum of squares to the square of the random sum with applications to risk measures." Publications de l'Institut Math?matique (Belgrade) 80, no. 94 (2006): 219–40. http://dx.doi.org/10.2298/pim0694219l.

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Let {X1,X2, . . .} be a sequence of independent and identically distributed positive random variables of Pareto-type and let {N(t); t >_ 0} be a counting process independent of the Xi?s. For any fixed t> _ 0, define: TN(t) := X2 1 + X2 2 + ? ? ? + X2N (t) (X1 + X2 + ? ? ? + XN(t))2 if N(t) >_ 1 and TN(t) := 0 otherwise. We derive limits in distribution for TN(t) under some convergence conditions on the counting process. This is even achieved when both the numerator and the denominator defining TN(t) exhibit an erratic behavior (EX1 = ?) or when only the numerator has an erratic behavi

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41

Hart, Eric W., and W. Gary Martin. "Standards for High School Mathematics: Why, What, How?" Mathematics Teacher 102, no. 5 (2008): 377–82. http://dx.doi.org/10.5951/mt.102.5.0377.

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Compare your own experience of learning algebra with Bertrand Russell's recollection: I was made to learn by heart: “The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.” I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way.”

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42

Hart, Eric W., and W. Gary Martin. "Standards for High School Mathematics: Why, What, How?" Mathematics Teacher 102, no. 5 (2008): 377–82. http://dx.doi.org/10.5951/mt.102.5.0377.

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Compare your own experience of learning algebra with Bertrand Russell's recollection: I was made to learn by heart: “The square of the sum of two numbers is equal to the sum of their squares increased by twice their product.” I had not the vaguest idea what this meant and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way.”

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43

Baker, R. C., and J. Brüdern. "On sums of two squarefull numbers." Mathematical Proceedings of the Cambridge Philosophical Society 116, no. 1 (1994): 1–5. http://dx.doi.org/10.1017/s0305004100072340.

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A natural number n is said to be squarefull if p|n implies p2|n for primes p. The set of all squarefull numbers is not much more dense in the natural numbers than the set of perfect squares but their additive properties may be rather different. We are more precise only in the case of sums of two such integers as this is the problem with which we are concerned here. Let U(x) be the number of integers not exceeding x and representable as the sum of two integer squares. Then, according to a theorem of Landau [4],as x tends to infinity.

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44

Goldoni, Giorgio. "A visual proof for the sum of the firstn squares and for the sum of the firstn factorials of order two." Mathematical Intelligencer 24, no. 4 (2002): 67–70. http://dx.doi.org/10.1007/bf03025326.

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45

Shiu, Peter. "The three-square theorem of Gauss and Legendre." Mathematical Gazette 104, no. 560 (2020): 209–14. http://dx.doi.org/10.1017/mag.2020.42.

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The following theorems are famous landmarks in the history of number theory.Theorem 1 (Fermat-Euler): A number is representable as a sum of two squares if, and only if, it has the form PQ2, where P is free of prime divisors q ≡ 3 (mod 4).Theorem 2 (Lagrange): Every number is representable as a sum of four squares.Theorem 3 (Gauss-Legendre): A number is representable as a sum of three squares if, and only if, it is not of the form 4a (8n + 7).

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46

Yiu, Paul Y. H. "Sums of Squares Formulae With Integer Coefficients." Canadian Mathematical Bulletin 30, no. 3 (1987): 318–24. http://dx.doi.org/10.4153/cmb-1987-045-6.

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AbstractHidden behind a sums of squares formula are other such formulae not obtainable by restriction. This drastically simplifies the combinatorics involved in the existence problem of sums of squares formulae, and leads to a proof that the product of two sums of 16 squares cannot be rewritten as a sum of 28 squares, if only integer coefficients are permitted. We also construct all [10, 10, 16] formulae.

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47

Rybalov, A. N. "On generic complexity of the problem to represent natural numbers by sum of two squares." Prikladnaya diskretnaya matematika. Prilozhenie, no. 13 (September 1, 2020): 111–13. http://dx.doi.org/10.17223/2226308x/13/33.

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48

Overmars, Anthony, and Sitalakshmi Venkatraman. "Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime." Mathematical and Computational Applications 25, no. 4 (2020): 63. http://dx.doi.org/10.3390/mca25040063.

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The security of RSA relies on the computationally challenging factorization of RSA modulus N=p1 p2 with N being a large semi-prime consisting of two primes p1and p2, for the generation of RSA keys in commonly adopted cryptosystems. The property of p1 and p2, both congruent to 1 mod 4, is used in Euler’s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied i

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49

Sun, Cui-Fang, and Zhi Cheng. "On the addition of two weighted squares of units mod n." International Journal of Number Theory 12, no. 07 (2016): 1783–90. http://dx.doi.org/10.1142/s1793042116501098.

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For any positive integer [Formula: see text], let [Formula: see text] be the ring of residue classes modulo [Formula: see text] and [Formula: see text] be the group of its units. Recently, for any [Formula: see text], Yang and Tang obtained a formula for the number of solutions of the quadratic congruence [Formula: see text] with [Formula: see text] units, nonunits and mixed pairs, respectively. In this paper, for any [Formula: see text], we give a formula for the number of representations of [Formula: see text] as the sum of two weighted squares of units modulo [Formula: see text]. We resolve

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50

López-Saldívar, Julio, Octavio Castaños, Eduardo Nahmad-Achar, Ramón López-Peña, Margarita Man’ko, and Vladimir Man’ko. "Geometry and Entanglement of Two-Qubit States in the Quantum Probabilistic Representation." Entropy 20, no. 9 (2018): 630. http://dx.doi.org/10.3390/e20090630.

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A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres–Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevic

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