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Journal articles on the topic 'Linear equations in primes'

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Published: 4 June 2021
Last updated: 6 February 2022

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1

Green, Benjamin, and Terence Tao. "Linear equations in primes." Annals of Mathematics 171, no. 3 (April 25, 2010): 1753–850. http://dx.doi.org/10.4007/annals.2010.171.1753.

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2

Balog, Antal. "Linear equations in primes." Mathematika 39, no. 2 (December 1992): 367–78. http://dx.doi.org/10.1112/s0025579300015096.

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3

Balog, Antal. "Six Primes and an Almost Prime in Four Linear Equations." Canadian Journal of Mathematics 50, no. 3 (June 1, 1998): 465–86. http://dx.doi.org/10.4153/cjm-1998-025-1.

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AbstractThere are infinitely many triplets of primes p, q, r such that the arithmetic means of any two of them, are also primes. We give an asymptotic formula for the number of such triplets up to a limit. The more involved problem of asking that in addition to the above the arithmetic mean of all three of them, is also prime seems to be out of reach. We show by combining the Hardy-Littlewood method with the sieve method that there are quite a few triplets for which six of the seven entries are primes and the last is almost prime.
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4

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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5

Zhixin Liu. "Small Prime Solutions to Cubic Diophantine Equations." Canadian Mathematical Bulletin 56, no. 4 (December 1, 2013): 785–94. http://dx.doi.org/10.4153/cmb-2012-025-0.

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Abstract.Let a1,;… a9 be nonzero integers and n any integer. Suppose that a1+…+a9 ≡ n (mod 2) and (ai ; aj ) = 1 for 1 ≤ i < j ≤9. In this paper we prove the following:(i) If aj are not all of the same sign, then the cubic equation has prime solutions satisfying pj ≪|n|1/3 + max{|aj|}14+∊.(ii) If all aj are positive and n ≫ max{|aj|} 43+∊, then is solvable in primes pj.These results are an extension of the linear and quadratic relative problems.
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6

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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7

Shparlinski, Igor E. "Evasive properties of sparse graphs and some linear equations in primes." Theoretical Computer Science 547 (August 2014): 117–21. http://dx.doi.org/10.1016/j.tcs.2014.06.005.

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8

KONG, YAFANG, and ZHIXIN LIU. "ON PAIRS OF GOLDBACH–LINNIK EQUATIONS." Bulletin of the Australian Mathematical Society 95, no. 2 (October 19, 2016): 199–208. http://dx.doi.org/10.1017/s000497271600071x.

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In this paper, we show that every pair of large positive even integers can be represented in the form of a pair of Goldbach–Linnik equations, that is, linear equations in two primes and $k$ powers of two. In particular, $k=34$ powers of two suffice, in general, and $k=18$ under the generalised Riemann hypothesis. Our result sharpens the number of powers of two in previous results, which gave $k=62$, in general, and $k=31$ under the generalised Riemann hypothesis.
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9

Bienvenu, Pierre-Yves. "Asymptotics for some polynomial patterns in the primes." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 5 (January 17, 2019): 1241–90. http://dx.doi.org/10.1017/prm.2018.52.

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AbstractWe prove asymptotic formulae for sums of the form $$\sum\limits_{n\in {\open z}^d\cap K} {\prod\limits_{i = 1}^t {F_i} } (\psi _i(n)),$$where K is a convex body, each Fi is either the von Mangoldt function or the representation function of a quadratic form, and Ψ = (ψ1, …, ψt) is a system of linear forms of finite complexity. When all the functions Fi are equal to the von Mangoldt function, we recover a result of Green and Tao, while when they are all representation functions of quadratic forms, we recover a result of Matthiesen. Our formulae imply asymptotics for some polynomial patterns in the primes. For instance, they describe the asymptotic behaviour of the number of k-term arithmetic progressions of primes whose common difference is a sum of two squares.The paper combines ingredients from the work of Green and Tao on linear equations in primes and that of Matthiesen on linear correlations amongst integers represented by a quadratic form. To make the von Mangoldt function compatible with the representation function of a quadratic form, we provide a new pseudorandom majorant for both – an average of the known majorants for each of the functions – and prove that it has the required pseudorandomness properties.
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10

Shaw, Sen-Yen. "Solvability of linear functional equations in Lebesgue spaces." Publications of the Research Institute for Mathematical Sciences 26, no. 4 (1990): 691–99. http://dx.doi.org/10.2977/prims/1195170854.

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11

Liu, Jianya, and Kai-Man Tsang. "Small prime solutions of ternary linear equations." Acta Arithmetica 118, no. 1 (2005): 79–100. http://dx.doi.org/10.4064/aa118-1-5.

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12

Li, Hongze. "Small prime solutions of linear ternary equations." Acta Arithmetica 98, no. 3 (2001): 293–309. http://dx.doi.org/10.4064/aa98-3-6.

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13

Ulmer, Felix. "Irreducible Linear Differential Equations of Prime Order." Journal of Symbolic Computation 18, no. 4 (October 1994): 385–401. http://dx.doi.org/10.1006/jsco.1994.1055.

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14

Meng, Xianmeng. "Linear Equations with Small Prime and Almost Prime Solutions." Canadian Mathematical Bulletin 51, no. 3 (September 1, 2008): 399–405. http://dx.doi.org/10.4153/cmb-2008-040-9.

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AbstractLet b1, b2 be any integers such that gcd(b1, b2) = 1 and c1|b1| < |b2| ≤ c2|b1|, where c1, c2 are any given positive constants. Let n be any integer satisfying gcd(n, bi) = 1, i = 1, 2. Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. In this paper, for almost all b2, we prove (i) a sharp lower bound for n such that the equation b1p + b2m = n is solvable in prime p and almost prime m = Pk, k ≥ 3 whenever both bi are positive, and (ii) a sharp upper bound for the least solutions p, m of the above equation whenever bi are not of the same sign, where p is a prime and m = Pk, k ≥ 3.
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15

Meng, Xianmeng. "On linear equations with prime variables of special type." Journal of Number Theory 129, no. 10 (October 2009): 2504–18. http://dx.doi.org/10.1016/j.jnt.2009.03.006.

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16

Ching, Tak Wing, and Kai Man Tsang. "Small prime solutions to linear equations in three variables." Acta Arithmetica 178, no. 1 (2017): 57–76. http://dx.doi.org/10.4064/aa8427-8-2016.

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17

Choi, Kwok-Kwong, Ming-Chit Liu, and Kai-Man Tsang. "Conditional bounds for small prime solutions of linear equations." Manuscripta Mathematica 74, no. 1 (December 1992): 321–40. http://dx.doi.org/10.1007/bf02567674.

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18

Li*, Hong Ze, and Richard J. Mcintosh**. "On Pairs of Linear Equations in Three Prime Variables." Acta Mathematica Sinica, English Series 20, no. 5 (June 21, 2004): 837–50. http://dx.doi.org/10.1007/s10114-003-0302-2.

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19

Hibino, Masaki. "Formal Gevrey Theory for Singular First Order Quasi-Linear Partial Differential Equations." Publications of the Research Institute for Mathematical Sciences 42, no. 4 (2006): 933–85. http://dx.doi.org/10.2977/prims/1166642193.

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20

Hibino, Masaki. "Divergence property of formal solutions for singular first order linear partial differential equations." Publications of the Research Institute for Mathematical Sciences 35, no. 6 (1999): 893–919. http://dx.doi.org/10.2977/prims/1195143361.

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21

Chen, Hua, and Hidetoshi Tahara. "On totally characteristic type non-linear partial differential equations in the complex domain." Publications of the Research Institute for Mathematical Sciences 35, no. 4 (1999): 621–36. http://dx.doi.org/10.2977/prims/1195143496.

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22

DERKSEN, H., and D. MASSER. "Linear equations over multiplicative groups, recurrences, and mixing III." Ergodic Theory and Dynamical Systems 38, no. 7 (May 2, 2017): 2625–43. http://dx.doi.org/10.1017/etds.2016.137.

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Given an algebraic $\mathbf{Z}^{d}$-action corresponding to a prime ideal of a Laurent ring of polynomials in several variables, we show how to find the smallest order $n+1$ of non-mixing. It is known that this is determined by the non-mixing sets of size $n+1$, and we show how to find these in an effective way. When the underlying characteristic is positive and $n\geq 2$, we prove that there are at most finitely many classes under a natural equivalence relation. We work out two examples, the first with five classes and the second with 134 classes.
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23

Cui, Zhen, and Hong Ze Li. "Small prime solutions of a pair of linear equations in five prime variables." Acta Mathematica Sinica, English Series 26, no. 3 (February 15, 2010): 569–78. http://dx.doi.org/10.1007/s10114-010-8031-9.

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24

Sasaki, Tateaki. "Cramer-type formula for the polynomial solutions of coupled linear equations with polynomial coefficients." Publications of the Research Institute for Mathematical Sciences 21, no. 1 (1985): 237–54. http://dx.doi.org/10.2977/prims/1195179845.

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25

Mochizuki, Kiyoshi, and Hideo Nakazawa. "Energy decay of solutions to the wave equations with linear dissipation localized near infinity." Publications of the Research Institute for Mathematical Sciences 37, no. 3 (2001): 441–58. http://dx.doi.org/10.2977/prims/1145477231.

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26

Ōuchi, Sunao. "Singular solutions with asymptotic expansion of linear partial differential equations in the complex domain." Publications of the Research Institute for Mathematical Sciences 34, no. 4 (1998): 291–311. http://dx.doi.org/10.2977/prims/1195144627.

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27

Mochizuki, Kiyoshi, and Hideo Nakazawa. "Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation." Publications of the Research Institute for Mathematical Sciences 32, no. 3 (1996): 401–14. http://dx.doi.org/10.2977/prims/1195162849.

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28

Erazo, Harold, Carlos A. Gómez, and Florian Luca. "Linear combinations of prime powers in X-coordinates of Pell equations." Ramanujan Journal 53, no. 1 (March 2, 2020): 123–37. http://dx.doi.org/10.1007/s11139-019-00213-5.

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29

Buşe, Constantin, Vasile Lupulescu, and Donal O'Regan. "Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 5 (March 20, 2019): 2175–88. http://dx.doi.org/10.1017/prm.2019.12.

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AbstractLetqbe a positive integer and let (an) and (bn) be two given ℂ-valued andq-periodic sequences. First we prove that the linear recurrence in ℂ0.1$$x_{n + 2} = a_nx_{n + 1} + b_nx_n,\quad n\in {\open Z}_+ $$is Hyers–Ulam stable if and only if the spectrum of the monodromy matrixTq: =Aq−1· · ·A0(i.e. the set of all its eigenvalues) does not intersect the unit circle Γ = {z∈ ℂ: |z| = 1}, i.e.Tqis hyperbolic. Here (and in as follows) we let0.2$$A_n = \left( {\matrix{ 0 & 1 \cr {b_n} & {a_n} \cr } } \right)\quad n\in {\open Z}_+ .$$Secondly we prove that the linear differential equation0.3$${x}^{\prime \prime}(t) = a(t){x}^{\prime}(t) + b(t)x(t),\quad t\in {\open R},$$(wherea(t) andb(t) are ℂ-valued continuous and 1-periodic functions defined on ℝ) is Hyers–Ulam stable if and only ifP(1) is hyperbolic; hereP(t) denotes the solution of the first-order matrix 2-dimensional differential system0.4$${X}^{\prime}(t) = A(t)X(t),\quad t\in {\open R},\quad X(0) = I_2,$$whereI2is the identity matrix of order 2 and0.5$$A(t) = \left( {\matrix{ 0 & 1 \cr {b(t)} & {a(t)} \cr } } \right),\quad t\in {\open R}.$$
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30

Dosoudilová, Monika, and Alexander Lomtatidze. "Remark on zeros of solutions of second-order linear ordinary differential equations." Georgian Mathematical Journal 23, no. 4 (December 1, 2016): 571–77. http://dx.doi.org/10.1515/gmj-2016-0052.

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AbstractAn efficient condition is established ensuring that on any interval of length ω, any nontrivial solution of the equation ${u^{\prime\prime}=p(t)u}$ has at most one zero. Based on this result, the unique solvability of a periodic boundary value problem is studied.
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31

Ōuchi, Sunao. "Existence of solutions with asymptotic expansion of linear partial differential equations in the complex domain." Publications of the Research Institute for Mathematical Sciences 40, no. 1 (2004): 239–94. http://dx.doi.org/10.2977/prims/1145475972.

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32

Hibino, Masaki. "Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type, II." Publications of the Research Institute for Mathematical Sciences 37, no. 4 (2001): 579–614. http://dx.doi.org/10.2977/prims/1145477330.

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33

Koike, Tatsuya. "On the exact WKB analysis of second order linear ordinary differential equations with simple poles." Publications of the Research Institute for Mathematical Sciences 36, no. 2 (2000): 297–319. http://dx.doi.org/10.2977/prims/1195143105.

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34

Liu, Ming-Chit, and Tianze Wang. "A numerical bound for small prime solutions of some ternary linear equations." Acta Arithmetica 86, no. 4 (1998): 343–83. http://dx.doi.org/10.4064/aa-86-4-343-383.

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35

Abraev, B., S. Nishonov, and I. Tilavov. "Special set of problems on solutions of linear equations in prime numbers." ACADEMICIA: An International Multidisciplinary Research Journal 10, no. 6 (2020): 870. http://dx.doi.org/10.5958/2249-7137.2020.00674.6.

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36

Opluštil, Zdeněk. "On non-oscillation for certain system of non-linear ordinary differential equations." Georgian Mathematical Journal 24, no. 2 (June 1, 2017): 277–85. http://dx.doi.org/10.1515/gmj-2016-0068.

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AbstractWe consider the following two-dimensional system of non-linear equations:u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u,where {\alpha>0}, and {g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and {p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on {[0,+\infty]}. We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.
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37

Ōuchi, Sunao. "An integral representation of singular solutions and removable singularities of solutions to linear partial differential equations." Publications of the Research Institute for Mathematical Sciences 26, no. 5 (1990): 735–83. http://dx.doi.org/10.2977/prims/1195170733.

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38

Demenchuk, A. K. "STRONGLY IRREGULAR PERIODIC SOLUTIONS OF THE FIRST-ORDER LINEAR HOMOGENEOUS DISCRETE EQUATION." Doklady of the National Academy of Sciences of Belarus 62, no. 3 (June 30, 2018): 263–67. http://dx.doi.org/10.29235/1561-8323-2018-62-3-263-267.

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In 1950 J. Massera proved that a fi rst-order scalar periodic ordinary differential equation has no strongly ira proved that a first-order scalar periodic ordinary differential equation has no strongly irregular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation. For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the above result of J. Massera has no complete analog.The purpose of this article is to investigate the possibility to realize Massera’s theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a first-order linear homogeneous non-stationary periodic discrete equation has no strongly irregular non-stationary periodic solutions.
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39

Choi, Stephen Kwok-Kwong, and Angel V. Kumchev. "Mean values of Dirichlet polynomials and applications to linear equations with prime variables." Acta Arithmetica 123, no. 2 (2006): 125–42. http://dx.doi.org/10.4064/aa123-2-2.

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40

KONG, YAFANG. "ON PAIRS OF LINEAR EQUATIONS IN FOUR PRIME VARIABLES AND POWERS OF TWO." Bulletin of the Australian Mathematical Society 87, no. 1 (March 22, 2012): 55–67. http://dx.doi.org/10.1017/s0004972712000172.

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AbstractIn this paper, we consider the simultaneous representation of pairs of positive integers. We show that every pair of large positive even integers can be represented in the form of a pair of linear equations in four prime variables and k powers of two. Here, k=63 in general and k=31 under the generalised Riemann hypothesis.
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41

Feng, Ruyong, Michael F. Singer, and Min Wu. "An algorithm to compute Liouvillian solutions of prime order linear difference–differential equations." Journal of Symbolic Computation 45, no. 3 (March 2010): 306–23. http://dx.doi.org/10.1016/j.jsc.2009.09.002.

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42

Liu, Ming-Chit, Tianze Wang, and Yuan Wang. "A numerical bound for small prime solutions of some ternary linear equations, II." Asian Journal of Mathematics 4, no. 4 (2000): 961–76. http://dx.doi.org/10.4310/ajm.2000.v4.n4.a13.

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43

Attia, E. R., V. Benekas, H. A. El-Morshedy, and I. P. Stavroulakis. "Oscillation of first order linear differential equations with several non-monotone delays." Open Mathematics 16, no. 1 (February 23, 2018): 83–94. http://dx.doi.org/10.1515/math-2018-0010.

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AbstractConsider the first-order linear differential equation with several retarded arguments$$\begin{array}{} \displaystyle x^{\prime }(t)+\sum\limits_{k=1}^{n}p_{k}(t)x(\tau _{k}(t))=0,\;\;\;t\geq t_{0}, \end{array} $$where the functions pk, τk ∈ C([t0, ∞), ℝ+), τk(t) < t for t ≥ t0 and limt→∞τk(t) = ∞, for every k = 1, 2, …, n. Oscillation conditions which essentially improve known results in the literature are established. An example illustrating the results is given.
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44

Ōuchi, Sunao. "Asymptotic expansion of singular solutions and the characteristic polygon of linear partial differential equations in the complex domain." Publications of the Research Institute for Mathematical Sciences 36, no. 4 (2000): 457–82. http://dx.doi.org/10.2977/prims/1195142869.

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45

Demenchuk, A. K. "On strongly irregular periodic solutions of the linear nonhomogeneous discrete equation of the first order." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 56, no. 1 (April 6, 2020): 30–35. http://dx.doi.org/10.29235/1561-2430-2020-56-1-30-35.

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As is proved earlier (the Massera theorem), the first-order scalar periodic ordinary differential equation does not have strongly irregular periodic solutions (solutions with a period incommensurable with the period of the equation). For difference equations with discrete time, strong irregularity means that the equation period and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the mentioned result has no complete analog.The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions.
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46

Bu, Shangquan, and Gang Cai. "Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces." Canadian Mathematical Bulletin 62, no. 4 (October 15, 2018): 715–26. http://dx.doi.org/10.4153/s0008439518000048.

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AbstractIn this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$-well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$, ($t\in \mathbb{R}$), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$, $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.
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47

Ōuchi, Sunao. "The behaviour of solutions with singularities on a characteristic surface to linear partial differential equations in the complex domains." Publications of the Research Institute for Mathematical Sciences 29, no. 1 (1993): 63–120. http://dx.doi.org/10.2977/prims/1195167544.

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48

Ozawa, Tohru, and Nakao Hayashi. "Lower bounds for order of decay or of growth in time for solutions to linear and nonlinear Schrödinger equations." Publications of the Research Institute for Mathematical Sciences 25, no. 6 (1989): 847–59. http://dx.doi.org/10.2977/prims/1195172508.

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49

LUCA, FLORIAN, and ANTONIN RIFFAUT. "LINEAR INDEPENDENCE OF POWERS OF SINGULAR MODULI OF DEGREE THREE." Bulletin of the Australian Mathematical Society 99, no. 1 (September 12, 2018): 42–50. http://dx.doi.org/10.1017/s0004972718000965.

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We show that two distinct singular moduli $j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers $m$ and $n$ the numbers $1,j(\unicode[STIX]{x1D70F})^{m}$ and $j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over $\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.
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50

HANKE, MARTIN, and ELISABETH RÖSLER. "COMPUTATION OF LOCAL VOLATILITIES FROM REGULARIZED DUPIRE EQUATIONS." International Journal of Theoretical and Applied Finance 08, no. 02 (March 2005): 207–21. http://dx.doi.org/10.1142/s0219024905002950.

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We propose a new method to calibrate the local volatility function of an asset from observed option prices of the underlying. Our method is initialized with a preprocessing step in which the given data are smoothened using cubic splines before they are differentiated numerically. In a second step the Dupire equation is rewritten as a linear equation for a rational expression of the local volatility. This equation is solved with Tikhonov regularization, using some discrete gradient approximation as penalty term. We show that this procedure yields local volatilities which appear to be qualitatively correct.
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