Journal articles: 'Diophantine equations'

1

Bruno, Alexander Dmitrievich. "From Diophantine approximations to Diophantine equations." Keldysh Institute Preprints, no. 1 (2016): 1–20. http://dx.doi.org/10.20948/prepr-2016-1.

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Aggarwal, S., and S. Kumar. "On the Exponential Diophantine Equation (13^2m) + (6r + 1)ⁿ = z²." Journal of Scientific Research 13, no. 3 (September 1, 2021): 845–49. http://dx.doi.org/10.3329/jsr.v13i3.52611.

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Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers. Results of the present paper show that the exponential Diophantine equation (132m) + (6r + 1)n = Z2, where m, n, r, z are whole numbers, has no solution in the whole number.

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3

Srinivasa Rao, K., T. S. Santhanam, and V. Rajeswari. "Multiplicative Diophantine equations." Journal of Number Theory 42, no. 1 (September 1992): 20–31. http://dx.doi.org/10.1016/0022-314x(92)90105-x.

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4

Choudhry, Ajai. "Symmetric Diophantine Equations." Rocky Mountain Journal of Mathematics 34, no. 4 (December 2004): 1281–98. http://dx.doi.org/10.1216/rmjm/1181069800.

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Hajdu, L., and Á. Pintér. "Combinatorial Diophantine equations." Publicationes Mathematicae Debrecen 56, no. 3-4 (April 1, 2000): 391–403. http://dx.doi.org/10.5486/pmd.2000.2179.

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6

Cohn, J. H. E. "Twelve diophantine equations." Archiv der Mathematik 65, no. 2 (August 1995): 130–33. http://dx.doi.org/10.1007/bf01270690.

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7

Tahiliani, Dr Sanjay. "More on Diophantine Equations." International Journal of Management and Humanities 5, no. 6 (February 28, 2021): 26–27. http://dx.doi.org/10.35940/ijmh.l1081.025621.

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In this paper, we will find the solutions of many Diophantine equations.Some are of the form 2(3 x )+ 5(7y ) +11=z2 for non negative x,y and z. we also investigate solutions ofthe Diophantine equation 2(x+3) +11(3y ) ─ 1= z2 for non negative x,y and z and finally, westudy the Diophantine equations (k!×k)n = (n!×n)k and ( 𝒌! 𝒌 ) 𝒏 = ( 𝒏! 𝒏 ) 𝒌 where k and n are positive integers. We show that the first one holds if and only if k=n and the second one holds if and only if k=n or (k,n) =(1,2),(2,1).We also investigate Diophantine equation u! + v! = uv and u! ─ v! = uv .

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8

Vălcan, Teodor Dumitru. "From Diophantian Equations to Matrix Equations (III) - Other Diophantian Quadratic Equations and Diophantian Equations of Higher Degree." Educatia 21, no. 25 (November 30, 2023): 167–77. http://dx.doi.org/10.24193/ed21.2023.25.18.

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In this paper, we propose to continue the steps started in the first two papers with the same generic title and symbolically denoted by (I) and (II), namely, the presentation of ways of achieving a systemic vision on a certain mathematical notional content, a vision that to motivate and mobilize the activity of those who teach in the classroom, thus facilitating both the teaching and the assimilation of notions, concepts, scientific theories approached by the educational disciplines that present phenomena and processes from nature. Thus, we will continue in the same systemic approach, solving some Diophantine equations of higher degree, more precisely some generalizations of the Pythagorean equation and some quadratic Diophantine equations, in the set of natural numbers, then of the whole numbers, in order to "submerge" a such an equation in a ring of matrices and try to find as many matrix solutions as possible. In this way we will solve 12 large classes of Diophantine quadratic or higher order equations. For attentive readers interested in these matters, at the end of the paper we will propose 6 open problems. The solution of each of these open problems represents, in fact, a vast research activity and that can open the way to solving such more complicated Diophantine and / or matrix equations.

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Acewicz, Marcin, and Karol Pąk. "Basic Diophantine Relations." Formalized Mathematics 26, no. 2 (July 1, 2018): 175–81. http://dx.doi.org/10.2478/forma-2018-0015.

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Summary The main purpose of formalization is to prove that two equations ya(z)= y, y = xz are Diophantine. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem. In our previous work [6], we showed that from the diophantine standpoint these equations can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities. In this formalization, we express these relations in terms of Diophantine set introduced in [7]. We prove that these relations are Diophantine and then we prove several second-order theorems that provide the ability to combine Diophantine relation using conjunctions and alternatives as well as to substitute the right-hand side of a given Diophantine equality as an argument in a given Diophantine relation. Finally, we investigate the possibilities of our approach to prove that the two equations, being the main purpose of this formalization, are Diophantine. The formalization by means of Mizar system [3], [2] follows Z. Adamowicz, P. Zbierski [1] as well as M. Davis [4].

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10

Biswas, D. "Does the Solution to the Non-linear Diophantine Equation 3x+35y=Z2 Exist?" Journal of Scientific Research 14, no. 3 (September 1, 2022): 861–65. http://dx.doi.org/10.3329/jsr.v14i3.58535.

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This paper investigates the solutions (if any) of the Diophantine equation 3x + 35y = Z2, where , x, y, and z are whole numbers. Diophantine equations are drawing the attention of researchers in diversified fields over the years. These are equations that have more unknowns than a number of equations. Diophantine equations are found in cryptography, chemistry, trigonometry, astronomy, and abstract algebra. The absence of any generalized method by which each Diophantine equation can be solved is a challenge for researchers. In the present communication, it is found with the help of congruence theory and Catalan’s conjecture that the Diophantine equation 3x + 35y = Z2 has only two solutions of (x, y, z) as (1, 0, 2) and (0, 1, 6) in non-negative integers.

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11

Wang, Min, Peng Yang, and Yining Yang. "Carlitz’s Equations on Generalized Fibonacci Numbers." Symmetry 14, no. 4 (April 7, 2022): 764. http://dx.doi.org/10.3390/sym14040764.

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Carlitz solved some Diophantine equations on Fibonacci or Lucas numbers. We extend his results to the sequence of generalized Fibonacci and Lucas numbers. In this paper, we solve the Diophantine equations of the form An1⋯Ank=Bm1⋯BmrCt1⋯Cts, where (An), (Bm), and (Ct) are generalized Fibonacci or Lucas numbers. Especially, we also find all solutions of symmetric Diophantine equations Ua1Ua2⋯Uam=Ub1Ub2⋯Ubn, where 1<a1≤a2≤⋯≤am, and 1<b1≤b2≤⋯≤bn.

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12

Alzer, Horst, and Florian Luca. "Diophantine equations involving factorials." Mathematica Bohemica 142, no. 2 (December 5, 2016): 181–84. http://dx.doi.org/10.21136/mb.2016.0045-15.

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13

Katayama, Shin-ichi, and Claude Levesque. "On simultaneous diophantine equations." Acta Arithmetica 108, no. 4 (2003): 369–77. http://dx.doi.org/10.4064/aa108-4-6.

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14

Babic, S. Bujacic, and K. Nabardi. "On some Diophantine equations." Miskolc Mathematical Notes 22, no. 1 (2021): 65. http://dx.doi.org/10.18514/mmn.2021.2638.

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15

Leitner, Dominik J. "Two exponential diophantine equations." Journal de Théorie des Nombres de Bordeaux 23, no. 2 (2011): 479–87. http://dx.doi.org/10.5802/jtnb.773.

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16

Brüdern, Jörg, and Rainer Dietmann. "Random Diophantine equations, I." Advances in Mathematics 256 (May 2014): 18–45. http://dx.doi.org/10.1016/j.aim.2014.01.017.

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17

Stroeker, Roelof J., and Benjamin M. M. de Weger. "Elliptic binomial diophantine equations." Mathematics of Computation 68, no. 227 (February 23, 1999): 1257–82. http://dx.doi.org/10.1090/s0025-5718-99-01047-9.

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18

Mollin, R. "Diophantine equations and congruences." International Journal of Algebra 1 (2007): 293–302. http://dx.doi.org/10.12988/ija.2007.07031.

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19

Jakimczuk, Rafael. "Diophantine equations. Elementary methods." International Mathematical Forum 12 (2017): 429–38. http://dx.doi.org/10.12988/imf.2017.7223.

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20

Baica, Malvina. "Diophantine equations and identities." International Journal of Mathematics and Mathematical Sciences 8, no. 4 (1985): 755–77. http://dx.doi.org/10.1155/s0161171285000849.

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The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are i) x2−my2=±1 ii) x3+my3+m2z3−3mxyz=1iii) Some fifth degree diopantine equationsInfinitely many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before.It is known that the solutions of Pell's equation are well exploited. We include it here because we shall use a common method to solve these three above mentioned equations and the method becomes very simple in Pell's equations case.Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields.

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21

Dofs, Erik. "Two exponential Diophantine equations." Glasgow Mathematical Journal 39, no. 2 (May 1997): 231–32. http://dx.doi.org/10.1017/s0017089500032122.

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In [3], two open problems were whether either of the diophantine equationswith n ∈ Z and f a prime number, is solvable if ω > 3 and 3 √ ω, but in this paper we allow f to be any (rational) integer and also 3 | ω. Equations of this form and more general ones can effectively be solved [5] with an advanced method based on analytical results, but the search limits are usually of enormous size. Here both equations (1) are norm equations in K (√–3): N(a + bp) = fw with p = (√–1 + –3)/2 which makes it possible to treat them arithmetically.

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22

Hering, Christoph. "On the Diophantine Equations." Applicable Algebra in Engineering, Communication and Computing 7, no. 4 (June 1, 1996): 251–62. http://dx.doi.org/10.1007/s002000050031.

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23

DĄBROWSKI, ANDRZEJ. "ON A CLASS OF GENERALIZED FERMAT EQUATIONS." Bulletin of the Australian Mathematical Society 82, no. 3 (June 18, 2010): 505–10. http://dx.doi.org/10.1017/s000497271000033x.

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AbstractWe generalize the main result of the paper by Bennett and Mulholland [‘On the diophantine equation xn+yn=2αpz2’, C. R. Math. Acad. Sci. Soc. R. Can.28 (2006), 6–11] concerning the solubility of the diophantine equation xn+yn=2αpz2. We also demonstrate, by way of examples, that questions about solubility of a class of diophantine equations of type (3,3,p) or (4,2,p) can be reduced, in certain cases, to studying several equations of the type (p,p,2).

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24

Maran, A. K. "A Simple Solution for Diophantine Equations of Second, Third and Fourth Power." Mapana - Journal of Sciences 4, no. 1 (September 12, 2005): 96–100. http://dx.doi.org/10.12723/mjs.6.17.

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We know already that the set Of positive integers, which are satisfying the Pythagoras equation Of three variables and four variables cre called Pythagorean triples and quadruples respectively. These cre Diophantine equation OF second power. The all unknowns in this Pythagorean equation have already Seen by mathematicians Euclid and Diophantine. Hcvwever the solution defined by Euclid are Diophantine is also again having unknowns. The only to solve the Diophantine equations wos and error method. Moreover, the trial and error method to obtain these values are not so practical and easy especially for time bound work, since the Diophantine equations are having more than unknown variables.

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25

Sankari, Hasan, and Mohammad Abobala. "On The Group of Units Classification In 3-Cyclic and 4-cyclic Refined Rings of Integers And The Proof of Von Shtawzens' Conjectures." International Journal of Neutrosophic Science 21, no. 4 (2023): 146–54. http://dx.doi.org/10.54216/ijns.210414.

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First Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with three variables . This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 6. Second Von Shtawzen's Diophantine equation is a non-linear Diophantine equation with four variables. This equation has been conjectured that it has a finite number of integer solutions, and this number of solutions is divisible by 8. In this paper, we prove that first Von Shtawzen's conjecture is true, where we show that first Von Shtawzen's Diophantine equations has exactly 12 solutions. On the other hand, we find all solutions of this Diophantine equations. In addition, we provide a full proof of second Von Shtawzen's conjecture, where we prove that the previous Diophantine equation has exactly 16 solutions, and we determine all of its possible solutions

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26

Choudhry, Ajai. "A new method of solving certain quartic and higher degree diophantine equations." International Journal of Number Theory 14, no. 08 (August 22, 2018): 2129–54. http://dx.doi.org/10.1142/s1793042118501282.

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In this paper, we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be applied to some diophantine systems in five or more variables. Under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, two examples being a sextic equation in four variables and two simultaneous equations of degrees four and six in six variables. We also simultaneously obtain arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. We obtain these solutions without finding a curve of genus 0 or 1 on the variety defined by the equations concerned. It appears that there exist projective varieties on which there are an arbitrarily large number of rational points and which do not contain a curve of genus 0 or 1 with infinitely many rational points.

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27

Chen, Imin. "A Diophantine Equation Associated to X0(5)." LMS Journal of Computation and Mathematics 8 (2005): 116–21. http://dx.doi.org/10.1112/s1461157000000929.

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AbstractSeveral classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat equations.

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28

Das, Radhika, Manju Somanath, and V. A. Bindu. "INTEGER SOLUTION ANALYSIS FOR A DIOPHANTINE EQUATION WITH EXPONENTIALS." jnanabha 53, no. 02 (2023): 69–73. http://dx.doi.org/10.58250/jnanabha.2023.53208.

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The exponential Diophantine equation is one of the distinctive types of Diophantine equations where the variables are expressed as exponents. For these equations, considerable excellent research has already been done. In this study, we try to solve the equations 3λ + 103μ = ξ2, 3λ + 181μ = ξ2, 3λ + 193μ = ξ2.

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29

Ulas, Maciej. "Some experiments with Ramanujan-Nagell type Diophantine equations." Glasnik Matematicki 49, no. 2 (December 18, 2014): 287–302. http://dx.doi.org/10.3336/gm.49.2.04.

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Anuradha Kameswari, P., S. S. Sriniasarao, and A. Belay. "AN APPLICATION OF LINEAR DIOPHANTINE EQUATIONS TO CRYPTOGRAPHY." Advances in Mathematics: Scientific Journal 10, no. 6 (June 10, 2021): 2799–806. http://dx.doi.org/10.37418/amsj.10.6.8.

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In this chapter we propose a Key exchange protocol based on a random solution of linear Diophantine equation in n variables, where the considered linear Diophantine equation satisfies the condition for existence of infinitely many solutions. Also the crypt analysis of the protocol is analysed.

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31

Wu, Yi, and Zheng Ping Zhang. "The Positive Integer Solutions of a Diophantine Equation." Applied Mechanics and Materials 713-715 (January 2015): 1483–86. http://dx.doi.org/10.4028/www.scientific.net/amm.713-715.1483.

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In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

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32

Vansan, Alexandre Hungaro. "EQUAÇÕES DIOFANTINAS: UM PROJETO PARA A SALA DE AULA E O USO DO GEOGEBRA." Ciência e Natura 37 (August 7, 2015): 532. http://dx.doi.org/10.5902/2179460x14629.

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http://dx.doi.org/10.5902/2179460X14629The study of Number Theory here in this article aims to study some properties of integer multiples or divisors, emphasizing issues related to divisibility, which will be of great importance for the study of Diophantine equations, which in turn will provide for applications using Geogebra software. The Diophantine equations are algebraic equations that show the solution set of integers, which in this paper we will discuss the Linear Diophantine equations with two unknowns of the form 𝑎 x + 𝑏 y = 𝑐 with 𝑎, 𝑏, 𝑐 integers. In which they are applied as an alternative way for students to find solutions to problems he faced during his school life. This work is intended to further training of teachers who are teaching in the elementary and high school, where you can find suggestions for activities that you can apply in the classroom, or even include in your lesson plan Diophantine equations, since here he will find a suggestion of teaching work plan to include in their classes.

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33

Kulkarni, Manisha, and B. Sury. "Diophantine equations with Bernoulli polynomials." Acta Arithmetica 116, no. 1 (2005): 25–34. http://dx.doi.org/10.4064/aa116-1-3.

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34

Grytczuk, Aleksander, and Izabela Kurzydlo. "On some matrix diophantine equations." Tsukuba Journal of Mathematics 33, no. 2 (December 2009): 299–304. http://dx.doi.org/10.21099/tkbjm/1267209422.

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35

AAAVoloch, José Felipe. "Commitment schemes and diophantine equations." Open Book Series 4, no. 1 (December 29, 2020): 1–5. http://dx.doi.org/10.2140/obs.2020.4.1.

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36

Kimura, Noriaki, and Kenneth S. Williams. "Infinitely Many Insolvable Diophantine Equations." American Mathematical Monthly 111, no. 10 (December 2004): 909. http://dx.doi.org/10.2307/4145100.

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37

Pocherevin, R. V. "Multidimensional system of Diophantine equations." Moscow University Mathematics Bulletin 72, no. 1 (January 2017): 41–43. http://dx.doi.org/10.3103/s0027132217010089.

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38

Brenner, J. L. "Book Review: Exponential Diophantine equations." Bulletin of the American Mathematical Society 25, no. 1 (July 1, 1991): 145–47. http://dx.doi.org/10.1090/s0273-0979-1991-16048-9.

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39

Magyar, Akos. "Diophantine equations and ergodic theorems." American Journal of Mathematics 124, no. 5 (2002): 921–53. http://dx.doi.org/10.1353/ajm.2002.0029.

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40

Kreso, Dijana, and Csaba Rakaczki. "Diophantine equations with Euler polynomials." Acta Arithmetica 161, no. 3 (2013): 267–81. http://dx.doi.org/10.4064/aa161-3-5.

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41

Buhanan, David, and Jaroslaw Kwapisz. "Cocyclic subshifts from Diophantine equations." Dynamical Systems 29, no. 1 (November 5, 2013): 56–66. http://dx.doi.org/10.1080/14689367.2013.844225.

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Kimura, Noriaki, and Kenneth S. Williams. "Infinitely Many Insolvable Diophantine Equations." American Mathematical Monthly 111, no. 10 (December 2004): 909–13. http://dx.doi.org/10.1080/00029890.2004.11920157.

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43

Berend, Daniel, and Jørgen E. Harmse. "On polynomial-factorial diophantine equations." Transactions of the American Mathematical Society 358, no. 4 (October 21, 2005): 1741–79. http://dx.doi.org/10.1090/s0002-9947-05-03780-3.

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44

Halter-Koch, Franz. "Diophantine equations of Pellian type." Journal of Number Theory 131, no. 9 (September 2011): 1597–615. http://dx.doi.org/10.1016/j.jnt.2011.02.005.

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45

Danilov, L. V. "Diophantine equations xm?Ayn=k." Mathematical Notes of the Academy of Sciences of the USSR 46, no. 6 (December 1989): 914–19. http://dx.doi.org/10.1007/bf01158625.

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46

Cook, Brian, and Ákos Magyar. "Diophantine equations in the primes." Inventiones mathematicae 198, no. 3 (April 23, 2014): 701–37. http://dx.doi.org/10.1007/s00222-014-0508-1.

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47

Mollin, R. A. "Diophantine equations and class numbers." Journal of Number Theory 24, no. 1 (September 1986): 7–19. http://dx.doi.org/10.1016/0022-314x(86)90053-3.

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48

Krasikov, I., and Y. Roditty. "Switching reconstruction and diophantine equations." Journal of Combinatorial Theory, Series B 54, no. 2 (March 1992): 189–95. http://dx.doi.org/10.1016/0095-8956(92)90050-8.

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49

Stephan Ramon Garcia, Vincent Selhorst-Jones, Daniel E. Poore, and Noah Simon. "Quotient Sets and Diophantine Equations." American Mathematical Monthly 118, no. 8 (2011): 704. http://dx.doi.org/10.4169/amer.math.monthly.118.08.704.

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50

Jakimczuk, Rafael. "Diophantine equations. Elementary methods II." International Mathematical Forum 12, no. 20 (2017): 953–65. http://dx.doi.org/10.12988/imf.2017.71192.

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Journal articles on the topic 'Diophantine equations'

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