Gotowe bibliografie tematyczne / Diophantine equations

Gotowa bibliografia na temat „Diophantine equations”

Autor: Grafiati
Data publikacji: 4 czerwca 2021
Data aktualizacji: 27 lipca 2024

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Artykuły w czasopismach na temat "Diophantine equations"

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Bruno, Alexander Dmitrievich. "From Diophantine approximations to Diophantine equations". Keldysh Institute Preprints, nr 1 (2016): 1–20. http://dx.doi.org/10.20948/prepr-2016-1.

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Aggarwal, S., i S. Kumar. "On the Exponential Diophantine Equation (132m) + (6r + 1)n = z2". Journal of Scientific Research 13, nr 3 (1.09.2021): 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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Srinivasa Rao, K., T. S. Santhanam i V. Rajeswari. "Multiplicative Diophantine equations". Journal of Number Theory 42, nr 1 (wrzesień 1992): 20–31. http://dx.doi.org/10.1016/0022-314x(92)90105-x.

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Choudhry, Ajai. "Symmetric Diophantine Equations". Rocky Mountain Journal of Mathematics 34, nr 4 (grudzień 2004): 1281–98. http://dx.doi.org/10.1216/rmjm/1181069800.

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Hajdu, L., i Á. Pintér. "Combinatorial Diophantine equations". Publicationes Mathematicae Debrecen 56, nr 3-4 (1.04.2000): 391–403. http://dx.doi.org/10.5486/pmd.2000.2179.

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Cohn, J. H. E. "Twelve diophantine equations". Archiv der Mathematik 65, nr 2 (sierpień 1995): 130–33. http://dx.doi.org/10.1007/bf01270690.

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Tahiliani, Dr Sanjay. "More on Diophantine Equations". International Journal of Management and Humanities 5, nr 6 (28.02.2021): 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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Vălcan, Teodor Dumitru. "From Diophantian Equations to Matrix Equations (III) - Other Diophantian Quadratic Equations and Diophantian Equations of Higher Degree". Educatia 21, nr 25 (30.11.2023): 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, i Karol Pąk. "Basic Diophantine Relations". Formalized Mathematics 26, nr 2 (1.07.2018): 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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Biswas, D. "Does the Solution to the Non-linear Diophantine Equation 3<sup>x</sup>+35<sup>y</sup>=Z<sup>2</sup> Exist?" Journal of Scientific Research 14, nr 3 (1.09.2022): 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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Rozprawy doktorskie na temat "Diophantine equations"

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Lee, Ellyn Jay. "Studies on diophantine equations". Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.239075.

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Mabaso, Automan Sibusiso. "Some exponential diophantine equations". Thesis, Stellenbosch : Stellenbosch University, 2013. http://hdl.handle.net/10019.1/85587.

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Thesis (MSc)--Stellenbosch University, 2013.
ENGLISH ABSTRACT: The aim of this thesis is to study some methods used in solving exponential Diophan- tine equations. There is no generic method or algorithm that can be used in solving all Diophantine equations. The main focus for our study will be solving the exponential Dio- phantine equations using the modular approach and the linear forms in two logarithms approach.
AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om sommige metodes te bestudeer om sekere Diophantiese vergelykings op te los. Daar is geen metode wat alle Diophantiese vergelykings kan oplos nie. Die fokus van os studie is hoofsaaklik om eksponensiele Diophantiese vergelykings op te los met die modul^ere metode en met die metode van line^ere vorms in twee logaritmes.
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Bartolomé, Boris. "Diophantine equations and cyclotomic fields". Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0104/document.

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Cette thèse examine quelques approches aux équations diophantiennes, en particulier les connexions entre l’analyse diophantienne et la théorie des corps cyclotomiques.Tout d’abord, nous proposons une introduction très sommaire et rapide aux méthodes d’analyse diophantienne que nous avons utilisées dans notre travail de recherche. Nous rappelons la notion de hauteur et présentons le PGCD logarithmique.Ensuite, nous attaquons une conjecture, formulée par Skolem en 1937, sur une équation diophantienne exponentielle. Pour cette conjecture, soit K un corps de nombres, α1 ,…, αm , λ1 ,…, λm des éléments non-nuls de K, et S un ensemble fini de places de K (qui contient toutes les places infinies), de telle sorte que l’anneau de S-entiers OS = OK,S = {α ∈ K : |α|v ≤ 1 pour les places v ∈/ S}contienne α1 , . . . , αm , λ1 , . . . , λm α1-1 , . . . , αm-1. Pour chaque n ∈ Z, soit A(n)=λ_1 α_1^n+⋯+λ_m α_m^n∈O_S. Skolem a suggéré [SK1] :Conjecture (principe local-global exponentiel). Supposons que pour chaque idéal non-nul a de l’anneau O_S, il existe n ∈ Z tel que A(n) ≡0 mod a. Alors, il existe n ∈ Z tel que A(n)=0.Soit Γ le groupe multiplicatif engendré par α1 ,…, αm. Alors Γ est le produit d’un groupe abélien fini et d’un groupe libre de rang fini. Nous démontrons que cette conjecture est vraie lorsque le rang de Γ est un.Après cela, nous généralisons un résultat précédent de Mourad Abouzaid ([A]). Soit F (X,Y) ∈ Q[X,Y] un Q-polynôme irréductible. En 2008, Mourad Abouzaid [A] a démontré le théorème suivant:Théorème (Abouzaid). Supposons que (0,0) soit un point non-singulier de la courbe plane F(X,Y) = 0. Soit m = degX F, n = degY F, M = max{m, n}. Soit ε tel que 0 < ε < 1. Alors, pour toute solution (α, β) ∈ Q ̅2 de F(X,Y) = 0, nous avons soit max{h(α), h(β)} ≤ 56M8ε−2hp(F) + 420M10ε−2 log(4M),soitmax{|h(α) − nlgcd(α, β)|,|h(β) − mlgcd(α, β)|} ≤ εmax{h(α), h(β)}++ 742M7ε−1hp(F) + 5762M9ε−1log(2m + 2n)Cependant, il a imposé la condition que (0,0) soit un point non-singulier de la courbe plane F(X,Y) = 0. En utilisant des versions quelque peu différentes du lemme “absolu” de Siegel et du lemme d’Eisenstein, nous avons pu lever la condition et démontrer le théorème de façon générale. Nous démontrons le théorème suivant:Théorème. Soit F(X,Y) ∈ Q ̅[X,Y] un polynôme absolument irréductible qui satisfasse F(0,0)=0. Soit m=degX F, n=degY F et r = min{i+j:(∂^(i+j) F)/(∂^i X∂^j Y)(0,0)≠0}. Soit ε tel que 0 < ε < 1. Alors, pour tout (α, β) ∈ Q ̅2 tel que F(α,β) = 0, nous avons soith(α) ≤ 200ε−2mn6(hp(F) + 5)soit|(lgcd(α,β))/r-h(α)/n|≤1/r (εh(α)+4000ε^(-1) n^4 (h_p (F)+log⁡(mn)+1)+30n^2 m(h_p (F)+log⁡(mn) ))Ensuite, nous donnons un aperçu des outils que nous avons utilisés dans les corps cyclotomiques. Nous tentons de développer une approche systématique pour un certain genre d’équations diophantiennes. Nous proposons quelques résultats sur les corps cyclotomiques, les anneaux de groupe et les sommes de Jacobi, qui nous seront utiles pour ensuite décrire l’approche.Finalement, nous développons une application de l’approche précédemment expliquée. Nous considèrerons l’équation diophantienne(1) Xn − 1 = BZn,où B ∈ Z est un paramètre. Définissons ϕ∗(B) := ϕ(rad (B)), où rad (B) est le radical de B, et supposons que(2) (n, ϕ∗(B)) = 1.Pour B ∈ N_(>1) fixé, soit N(B) = {n ∈ N_(>1) | ∃ k > 0 tel que n|ϕ∗(B)}. Si p est un premier impair, nous appellerons CF les conditions combinéesI La conjecture de Vandiver est vraie pour p, c’est-à-dire que le nombre de classe h+ du sous-corps réel maximal du corps cyclotomique Q[ζp ], n’est pas divisible par p.II Nous avons ir(p) < √p − 1, en d’autre mots, il y a au plus √p − 1 entiers impairs k < p tels que le nombre de Bernouilli Bk ≡ 0 mod p. [...]
This thesis examines some approaches to address Diophantine equations, specifically we focus on the connection between the Diophantine analysis and the theory of cyclotomic fields.First, we propose a quick introduction to the methods of Diophantine approximation we have used in this research work. We remind the notion of height and introduce the logarithmic gcd.Then, we address a conjecture, made by Thoralf Skolem in 1937, on an exponential Diophantine equation. For this conjecture, let K be a number field, α1 ,…, αm , λ1 ,…, λm non-zero elements in K, and S a finite set of places of K (containing all the infinite places) such that the ring of S-integersOS = OK,S = {α ∈ K : |α|v ≤ 1 pour les places v ∈/ S}contains α1 , . . . , αm , λ1 , . . . , λm α1-1 , . . . , αm-1. For each n ∈ Z, let A(n)=λ_1 α_1^n+⋯+λ_m α_m^n∈O_S. Skolem suggested [SK1] :Conjecture (exponential local-global principle). Assume that for every non zero ideal a of the ring O_S, there exists n ∈ Z such that A(n) ≡0 mod a. Then, there exists n ∈ Z such that A(n)=0.Let Γ be the multiplicative group generated by α1 ,…, αm. Then Γ is the product of a finite abelian group and a free abelian group of finite rank. We prove that the conjecture is true when the rank of Γ is one.After that, we generalize a result previously published by Abouzaid ([A]). Let F(X,Y) ∈ Q[X,Y] be an irreducible Q-polynomial. In 2008, Abouzaid [A] proved the following theorem:Theorem (Abouzaid). Assume that (0,0) is a non-singular point of the plane curve F(X,Y) = 0. Let m = degX F, n = degY F, M = max{m, n}. Let ε satisfy 0 < ε < 1. Then for any solution (α,β) ∈ Q ̅2 of F(X,Y) = 0, we have eithermax{h(α), h(β)} ≤ 56M8ε−2hp(F) + 420M10ε−2 log(4M),ormax{|h(α) − nlgcd(α, β)|,|h(β) − mlgcd(α, β)|} ≤ εmax{h(α), h(β)}++ 742M7ε−1hp(F) + 5762M9ε−1log(2m + 2n)However, he imposed the condition that (0, 0) be a non-singular point of the plane curve F(X,Y) = 0. Using a somewhat different version of Siegel’s “absolute” lemma and of Eisenstein’s lemma, we could remove the condition and prove it in full generality. We prove the following theorem:Theorem. Let F(X,Y) ∈ Q ̅[X,Y] be an absolutely irreducible polynomial satisfying F(0,0)=0. Let m=degX F, n=degY F and r = min{i+j:(∂^(i+j) F)/(∂^i X∂^j Y)(0,0)≠0}. Let ε be such that 0 < ε < 1. Then, for all (α, β) ∈ Q ̅2 such that F(α,β) = 0, we have eitherh(α) ≤ 200ε−2mn6(hp(F) + 5)or|(lgcd(α,β))/r-h(α)/n|≤1/r (εh(α)+4000ε^(-1) n^4 (h_p (F)+log⁡(mn)+1)+30n^2 m(h_p (F)+log⁡(mn) ))Then, we give an overview of the tools we have used in cyclotomic fields. We try there to develop a systematic approach to address a certain type of Diophantine equations. We discuss on cyclotomic extensions and give some basic but useful properties, on group-ring properties and on Jacobi sums.Finally, we show a very interesting application of the approach developed in the previous chapter. There, we consider the Diophantine equation(1) Xn − 1 = BZn,where B ∈ Z is understood as a parameter. Define ϕ∗(B) := ϕ(rad (B)), where rad (B) is the radical of B, and assume that (2) (n, ϕ∗(B)) = 1.For a fixed B ∈ N_(>1)we let N(B) = {n ∈ N_(>1) | ∃ k > 0 such that n|ϕ∗(B)}. If p is an odd prime, we shall denote by CF the combined condition requiring thatI The Vandiver Conjecture holds for p, so the class number h+ of the maximal real subfield of the cyclotomic field Q[ζp ] is not divisible by p.II We have ir>(p) < √p − 1, in other words, there is at most √p − 1 odd integers k < p such that the Bernoulli number Bk ≡ 0 mod p. [...]
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Yesilyurt, Deniz. "Solving Linear Diophantine Equations And Linear Congruential Equations". Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-19247.

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This report represents GCD, euclidean algorithm, linear diophantine equation and linear congruential equation. It investigates the methods for solving linear diophantine equations and linear congruential equations in several variables. There are many examples which illustrate the methods for solving equations.
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Smart, Nigel Paul. "The computer solution of diophantine equations". Thesis, University of Kent, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.315089.

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Long, Rachel Louise. "The algorithmic solution of simultaneous diophantine equations". Thesis, Oxford Brookes University, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444340.

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Baczkowski, Daniel M. "Diophantine Equations Involving Arithmetic Functions of Factorials". Miami University / OhioLINK, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=miami1088086258.

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Akhtari, Shabnam. "Thue equations and related topics". Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/1288.

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Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas.
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Schindler, Damaris. "On diophantine equations involving norm forms and bihomogeneous forms". Thesis, University of Bristol, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.601162.

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If the descent theory, developed by Colliot-Thelene and Sansuc, applies, then it can reduce the question of understanding whether the Brauer-Manin obstruction is the only one to understanding weak approximation on the resulting descent varieties. In some cases the descent varieties are easier to handle and accessible by analytic methods as for example t he circle method. In joint work with A. Skorobogatov we followed this approach focusing on varieties corresponding to the representation of a norm form by a product of linear polynomials. We present this work in the first part of this thesis which involves an application of the circle method over number fields to systems of linear equations involving norm forms. In the second part of this thesis we study the arithmetic of subvarieties in biprojective space. So far, the circle method has been a very useful tool to prove many cases of Manin's conjecture. Work of B. Birch back in 1962 establishes this for smooth complete intersections in projective space as soon as the number of variables is large enough depending on the degree and number of equations. In biprojective space there is not much known so far, unless the underlying polynomials are of bidegree (1,1). A combination of the circle method with the generalised hyperbola method recently developed by V. Blomer and J. Brudern allows us to verify Manin 's conjecture for certain smooth hypersurfaces of general bidegree in biprojective space.
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Ibrahim, Mostafa. "Modular and reciprocity approaches to a family of diophantine equations". Thesis, University of Warwick, 2009. http://wrap.warwick.ac.uk/2761/.

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In this thesis we study the Diophantine equation xp - Dy2p = z2; gcd(x; z) = 1; p prime: We combine two approaches: - The modular approach using in Wiles's proof of Fermat's Last Theorem. - Elementary quadratic reciprocity. We show how using this combination of approaches and computer calculations we can get congruence conditions for the exponent p.
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Książki na temat "Diophantine equations"

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Schmidt, Wolfgang M. Diophantine Approximations and Diophantine Equations. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0098246.

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Schmidt, Wolfgang M. Diophantine approximations and diophantine equations. Berlin: Springer, 1996.

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Diophantine approximations and diophantine equations. Berlin: Springer-Verlag, 1991.

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Andreescu, Titu, i Dorin Andrica. Quadratic Diophantine Equations. New York, NY: Springer New York, 2015. http://dx.doi.org/10.1007/978-0-387-54109-9.

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Sprindžuk, Vladimir G. Classical Diophantine Equations. Redaktor Ross Talent. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993. http://dx.doi.org/10.1007/bfb0073786.

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R, Tijdeman, red. Exponential diophantine equations. Cambridge [Cambridgeshire]: Cambridge University Press, 1986.

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Shorey, T. N. Exponential diophantine equations. Cambridge: Cambridge University Press, 2008.

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Sprindzhuk, V. G. Classical diophantine equations. Berlin: Springer-Verlag, 1993.

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1955-, Silverman Joseph H., red. Diophantus and diophantine equations. [Washington, DC]: Mathematical Association of America, 1997.

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Weger, B. M. M. Algorithms for diophantine equations. Amsterdam: Stichting Mathematisch Centrum, 1989.

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Części książek na temat "Diophantine equations"

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Ireland, Kenneth, i Michael Rosen. "Diophantine Equations". W A Classical Introduction to Modern Number Theory, 269–96. New York, NY: Springer New York, 1990. http://dx.doi.org/10.1007/978-1-4757-2103-4_17.

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Andreescu, Titu, i Dorin Andrica. "Diophantine Equations". W Number Theory, 1–17. Boston: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/b11856_19.

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Andreescu, Titu, i Dorin Andrica. "Diophantine Equations". W Number Theory, 1–21. Boston: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/b11856_8.

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Caminha Muniz Neto, Antonio. "Diophantine Equations". W Problem Books in Mathematics, 193–207. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-77977-5_7.

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Faltings, Gerd. "Diophantine Equations". W Mathematics Unlimited — 2001 and Beyond, 449–54. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-642-56478-9_21.

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Guy, Richard K. "Diophantine Equations". W Unsolved Problems in Number Theory, 139–98. New York, NY: Springer New York, 1994. http://dx.doi.org/10.1007/978-1-4899-3585-4_5.

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Schroeder, Manfred R. "Diophantine Equations". W Number Theory in Science and Communication, 95–110. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/978-3-662-22246-1_7.

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Guy, Richard K. "Diophantine Equations". W Unsolved Problems in Number Theory, 209–310. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/978-0-387-26677-0_5.

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Schroeder, Manfred R. "Diophantine Equations". W Number Theory in Science and Communication, 102–17. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-662-03430-9_7.

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Effinger, Gove, i Gary L. Mullen. "Diophantine Equations". W Elementary Number Theory, 139–64. Boca Raton: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003193111-9.

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Streszczenia konferencji na temat "Diophantine equations"

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Riemel, Tomáš. "SPECIAL EXPONENTIAL DIOPHANTINE EQUATIONS". W 15th annual International Conference of Education, Research and Innovation. IATED, 2022. http://dx.doi.org/10.21125/iceri.2022.0270.

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ȚARĂLUNGĂ, Boris. "About solutions of some non – linear Diophantine equations". W Ştiință și educație: noi abordări și perspective. "Ion Creanga" State Pedagogical University, 2023. http://dx.doi.org/10.46727/c.v3.24-25-03-2023.p293-298.

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In this paper, it is show that the Diophantine exponential equation:2^x +12^y=z^2 has exactly three non – negative integer solutions {(3,0,3),(2,1,4),(8,2,20)}, the Diophantine exponential equation: 2^x+14^y=z^2 has exactly three non–negative integer solutions: {(3,0,3), (1,1,4),(7,2,18)}, the Diophantine exponential equation: 2^x+15^y=z^2 has exactly two non–negative integer solutions: {(3,0,3), (6,2,17)}.
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Hu, Cheng-Feng, i Fung-Bao Liu. "Solving System of Fuzzy Diophantine Equations". W 2008 Fifth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD). IEEE, 2008. http://dx.doi.org/10.1109/fskd.2008.111.

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Avdyev, M. "THE DIOPHANTINE EQUATION FROM THE EYE OF PHYSICIST". W X Международная научно-практическая конференция "Культура, наука, образование: проблемы и перспективы". Нижневартовский государственный университет, 2022. http://dx.doi.org/10.36906/ksp-2022/57.

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A Diophantine equation is an equation with integer coefficients, the solutions of which must be found among integers. The equation is named after the mathematician Diophantus of Alexandria (III century). Despite its simplicity, a Diophantine equation may have a nontrivial solution (several solutions) or has no solution at all. Fermat's Last Theorem and Pythagorean Theorem are the Diophantine equations too. In 1900 The German mathematician David Hilbert formulated the Tenth problem. After 70 years, the answer turned out to be negative: there is no general algorithm. Nevertheless, for some cases, schoolchildren can understand whether a Diophantine equation is solvable without resorting to calculations, relying on the methods of physics, symmetry and set theory.
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He Kong, Bin Zhou i Mao-Rui Zhang. "A Stein equation approach for solutions to the Diophantine equations". W 2010 Chinese Control and Decision Conference (CCDC). IEEE, 2010. http://dx.doi.org/10.1109/ccdc.2010.5498658.

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Cipu, Mihai. "Groebner Bases and Solutions to Diophantine Equations". W 2008 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. IEEE, 2008. http://dx.doi.org/10.1109/synasc.2008.13.

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Walsh, P. G., i Takao Komatsu. "Recent Progress on Certain Quartic Diophantine Equations". W DIOPHANTINE ANALYSIS AND RELATED FIELDS: DARF 2007/2008. AIP, 2008. http://dx.doi.org/10.1063/1.2841908.

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Berkholz, Christoph, i Martin Grohe. "Linear Diophantine Equations, Group CSPs, and Graph Isomorphism". W Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2017. http://dx.doi.org/10.1137/1.9781611974782.21.

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Miyazaki, Takafumi, Masaaki Amou i Masanori Katsurada. "Exceptional cases of Terai’s conjecture on Diophantine equations". W DIOPHANTINE ANALYSIS AND RELATED FIELDS 2011: DARF - 2011. AIP, 2011. http://dx.doi.org/10.1063/1.3630043.

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Dhurga, C. Kanaga. "Reflecting the application of Diophantine equations in chemistry". W 2ND INTERNATIONAL CONFERENCE ON MATHEMATICAL TECHNIQUES AND APPLICATIONS: ICMTA2021. AIP Publishing, 2022. http://dx.doi.org/10.1063/5.0108562.

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Raporty organizacyjne na temat "Diophantine equations"

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Jain, Himanshu, Edmund M. Clarke i Orna Grumberg. Efficient Craig Interpolation for Linear Diophantine (Dis)Equations and Linear Modular Equations. Fort Belvoir, VA: Defense Technical Information Center, luty 2008. http://dx.doi.org/10.21236/ada476801.

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Osipov, Gennadij Sergeevich, Natella Semenovna Vashakidze i Galina Viktorovna Filippova. Fundamentals of solving linear Diophantine equations with two unknowns. Постулат, 2018. http://dx.doi.org/10.18411/postulat-2018-2-37.

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