Diophantine equations

Every second year, the GEOCRYPT conference brings together researchers from arithmetic geometry and cryptography. After Guadeloupe in 2009 and Corsica in 2011, French Polynesia was the host of its 2013 edition which took place in Punaauia, on the island of Tahiti, October 7-11. The main topic of GEOCRYPT has always been the application of pure mathematical techniques to the safety and efficiency of modern communication systems, with particular interest in the fields of arithmetic and algebraic geometry over finite fields, algorithms for finite fields, error correcting codes, cryptology, boolean functions, discrete dynamical systems, and their interactions. The 2013 edition had the honour to feature six invited and eleven contributed talks from renowned international experts presenting strong results recently obtained on topics ranging from pure mathematics to cryptographic algorithms. The GEOCRYPT 2013 Program Committee carefully evaluated the submitted abstracts and selected the best contributions for presentation at the conference.

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups [Formula: see text]. Then, we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups [Formula: see text].

We present the mathematical model of Information security system based on the linear inhomogeneous Diophantine equation. Plain text is the solution of the Diophantine equation, cipher text is the right side of equation. We also present the method of finding this solution. It is based on construction of a system of equations the solution of which is equal to desired solution of the original Diophantine equation. The system of equations is constructed using some secret information.Cryptanalysis of described mathematical model demonstrates the potential of using Diophantine equations for the development of Information security systems despite the existing vulnerabilities. The use the Diophantine equations allows to construct the systems which have a large variety of equally probable keys. And only one key is correct.

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.

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

In this paper we use certain properties of rational binary forms to solve several diophantine equations of the type f(x, y) = f(u, v). If on applying the nonsingular linear transformation T defined by x = αu + βv, y = γu + δv, the binary form φ(x, y) becomes a scalar multiple of the form φ(u, v), we call φ(x, y) an eigenform of the linear transformation T . If f(x, y) = L(x, y)φ(x, y) where φ(x, y) is an eigenform of the linear transformation T and L(x, y) is not an eigenform of T , the diophantine equation f(x, y) = f(u, v) reduces, on making the substitution x = m(αu+βv), y = m(γu+ δv), to a linear equation in the variables u and v while m is an arbitrary parameter. The solution of this linear equation readily yields a parametric solution of the original diophantine equation. We first use eigenforms to obtain parametric solutions of several general types of diophantine equations such as L1(x, y)Q1(x, y)Q s 2(x, y) = L1(u, v)Q r 1(u, v)Q s 2(u, v) and {Πi=1Li(x, y, z)}Qr(x, y, z) = {Πi=1Li(u, v,w)}Qr(u, v,w) where Ls and Qs denote linear and quadratic forms and r and s are arbitrary integers, and then we obtain parametric solutions of several specific diophantine equations such as the equation f(x, y) = f(u, v) where f(x, y) = xn + xn−1y + · · · + yn, n being an arbitrary odd integer and the equation x7 + y7 + 625z7 = u7 + v7 + 625w7.

The Undecidability of Exponential Diophantine Equations

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-integers OS = 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 either max{h(α), h(β)} ≤ 56M8ε−2hp(F) + 420M10ε−2 log(4M), or max{|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 either h(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 that I 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. Current results on Equation (1) are restricted to values of B which are built up from two small primes p ≤ 13 [BGMP] and complete solutions for B < 235 ([BBGP]). If expecting that the equation has no solutions, – possibly with the exception of some isolated examples – it is natural to consider the case when the exponent n is a prime. Of course, the existence of solutions (X,Z) for composite n imply the existence of some solutions with n prime, by raising X, Z to a power. The main contribution of our work has been to relate (1) in the case when n is a prime and (2) holds, to the diagonal Nagell – Ljunggren equation (X^n-1)/(X-1)=n^e Y^n, e={(O si X≢1[n]@1 sinon)┤ This way, we can apply results from [M] and prove the following: Theorem. Let n be a prime and B > 1 an integer with (ϕ∗(B), n) = 1. Suppose that equation (1) has a non-trivial integer solution different from n = 3 and (X,Z;B) = (18,7;17). Let X ≡ u mod n, 0 ≤ u < n and e = 1 if u = 1 and e = 0 otherwise. Then: 1. n > 163106. 2. X − 1 = ±B/ne and B < nn. 3. If u ∉ {−1,0,1}, then condition CF (II) fails for n and 2n−1 ≡ 3n−1 ≡ 1 mod n2 , and rn−1 ≡ 1 mod n2 for all r|X(X2 − 1). If u ∈ {−1, 0, 1}, then Condition CF (I) fails for n. Based on this theorem, we also prove the following: Theorem. If equation (1) has a solution for a fixed B verifying the conditions (2), then either n ∈ N(B) or there is a prime p coprime to ϕ∗(B) and a m ∈ N(B) such that n = p.m. Moreover, Xm, Ym are a solution of (1) for the prime exponent p and thus verify the conditions of the previous Theorem. This is a strong improvement of the currently known results. As we have made heavy use of [M], at the end of this thesis we have added an appendix to expose some new result that allows for a full justification of Theorem 3 of [M]. Keywords Diophantine Equations, Cyclotomic Fields, Nagell-Ljunggren Equation, Skolem, Abouzaid, Exponential Diophantine Equation, Baker’s Inequality, Subspace Theorem. References [A] M. Abouzaid, Heights and logarithmic gcd on algebraic curves, Int. J. Number Th. 4, pp. 177–197 (2008). [BBGP] A.Bazso, A.Bérczesc K.Györy and A.Pintér, On the resolution of equations Axn − Byn = C in integers x, y and n ≥ 3, II, Publicationes Mathematicae Debrecen 76, pp. 227 – 250 (2010). [BGMP] M. A. Bennett, K. Györy, M. Mignotte and A. Pintér, Binomial Thue equations and polynomial powers, Compositio Math. 142, pp. 1103–1121 (2006). [M] P. Mihăilescu Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren, Diophantine Approximation, Springer Verlag, Development in Mathematics 16, pp. 245–273 (2008). [S1] T. Skolem, Anwendung exponentieller Kongruenzen zum Beweis der Unlosbarkeit gewisser diophantischer Gleichungen, Avhdl. Norske Vid. Akad. Oslo I 12, pp. 1–16 (1929). [S2] T. Skolem, Lösung gewisser Gleichungssysteme in ganzen Zahlen oder ganzzahligen Polynomen mit beschränktem gemeinschaftlichen Teiler, Oslo Vid. Akar. Skr. I, 12 (1929).

We define a computable function f from positive integers to positive integers. We formulate a hypothesis which states that if a system S of equations of the forms x i · x j = x k and x i + 1 = x i has only finitely many solutions in non-negative integers x 1 , . . . , x i , then the solutions of S are bounded from above by f (2n). We prove the following: (1) the hypothesis implies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (3) the hypothesis implies that the question of whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (4) the hypothesis implies that if a set M ⊆ N has a finite-fold Diophantine representation, thenMis computable.

This study aims to highlight the importance of knowing the methods of solving Diophantine equations. The material is structured into: Introduction, Classes of Diophantine equations, presentation of first-degree Diophantine equations, Pythagorean triples and higher - Diophantine equations, methods for solving Diophantine equations. The paper describes and exemplifies different methods such as the decomposition method, the parametric method for solving Diophantine equations, solving Diophantine equations with inequalities through the method of modular arithmetic, mathematical induction, Fermat's method of infinite descent. Solving problems is illustrated by various applications of the mathematical results methods presented above. Any education, including mathematical education, has a double effect. On the one hand, the learner gains knowledge, on the other hand, he builds those skills which are engaged in work, develop the abilities needed to perform this education. Mathematical education builds thought. Of course, other actions are involved in building thought, but the role of Mathematical education is essential. This article is part of an empirical research on the teaching and learning of mathematics, teaching practices related to the main classes of Diophantine equations, leading to the development of cognitive skills in students.

A significant and important subject area of Theory of Numbers is the theory of Diophantine equations which concentrates on attempting to determine solutions in integers for higher degree and many parameters indeterminate equations. Obviously, polynomial Diophantine equations are many due to definition. Especially, the third degree Diophantine equation in two parameters falls into the theory of elliptic curves which is a developed theory. There are numerous motivating cubic equations with multiple variables which have kindled the interest among Mathematicians. For example, the representation of integers by binary cubic forms is known very little. In this context, for simplicity and brevity, refer various forms of equations of degree three having many variables in [Carmichael.,1959, Dickson.,1952, Mordell.,1969, Gopalan et.al., 2015a, Gopalan et.al., 2015b, Premalatha, Gopalan et., 2020, Premalatha et.al., 2021, Shanthi, Gopalan.,2023, Thiruniraiselvi, Gopalan.,2021, Thiruniraiselvi, Gopalan., 2024a, Thiruniraiselvi, Gopalan., 2024b, Thiruniraiselvi et.al., 2024 Vidhyalakshmi, Gopalan., 2022a, Vidhyalakshmi, Gopalan., 2022b, Vidhyalakshmi, Gopalan., 2022c]. The focus in this book is on solving multivariable third degree Diophantine equations. These types of equations are significant since they concentrate on obtaining solutions in integers which satisfy the considered algebraic equations. These solutions play a vital role in different area of mathematics &amp; science and help us in understanding the significance of number patterns. This book contains a reasonable collection of cubic Diophantine equations with three, four, five and six unknowns. The procedure in obtaining varieties of solutions in integers for the polynomial Diophantine equations of degree three with three , four, five and six unknowns considered in this book are illustrated in an elegant manner.

In this paper there are theorems that demonstrate the validity of using Diophantine equations parametric solutions properties for information security system mathematical models. For this the proof of a generalization of the known Frolov's theorem has been done and represented. Also we present new theorem as a mathematical model of information security system containing Diophantine problems.On the basis of two particular solutions a new method of parameterization of multigrade systems of Diophantine equations has been invented and presented in this paper. Particulary, on the basis of two equations with less variables this method allows to get general parametric solutions for multigrade Diophantine equations. On the example of the fifth degree equation, the parametric solutions of the multigrade system of Diophantine equations have been used as a mathematical model of a new cryptosystem.The new approach has been proposed for the development of information security system. It generalizes the principle of construction public key cryptosystems: one part of the conditional identity is used for the direct transformation of an original message, and other part is used for the inverse transformation.The represented mathematical models demonstrate the potential of using Diophantine equations for the development of information security system with a high degree of reliability. These models give an ability to build both a symmetrical system and an public key system. Such systems allow an existence of a countable set of equally probable keys that leads to Diophantine problems.

The theory of Diophantine equation offers a rich variety of fascinating problems. There are Diophantine problems, which involve cubic equations with four variables. The cubic Diophantine equation given by is analyzed for its patterns of non-zero distinct integral solutions. Objectives: The objective of this paper is to explore the integral solutions of cubic equation by using suitable methodologies. A few interesting relations between the solutions and special numbers are exhibited. Method: Solving Diophantine equation is obtained by the method of Decomposition. The structure of decomposition: like , where and Z. By the decomposing method in primary terms of a, we achieve a countable number of decompositions in k full factors . Each decomposition of this kind leads to a system of equations similar to: , . We get multitude of solutions for a given equation, by determining the system of equations. Findings: By the method of linear transformations, the ternary cubic equation with four unknowns is solved for its integral solutions. The equation is researched for its attributes and correlation among the solutions for its non – zero unique integer points. In each of the transformations taken, the cubic equation yields different solutions. The properties of the solutions and their relationship with the special numbers are also exhibited. Novelty: Mathematician’s interest towards solving Pell’s equation has been so much not because they approximate with a value for . The main importance of the Pell’s equation is due to that most of the common questions have answers in this equation which can be sorted by 2 variables in the Quadratic equations. This document is about the research on higher degree Cubic Diophantine equation which gives the integral solutions of this equation, taken into consideration. Keywords: Integral solutions, Ternary Cubic, Oblong number, Polygonal number

Article Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1 was published on May 29, 2001 in the journal Journal für die reine und angewandte Mathematik (volume 2001, issue 535).

Stiller proved that the Diophantine equation $x^2+119=15\cdot 2^{n}$ has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type $x^2=Ak^{n}+B$ with many solutions. Here, $A,B\in\Z$ (thus $A, B$ are not necessarily positive) and $k\in\Z_{\geq 2}$ are given integers. In particular, we prove that for each $k$ there exists an infinite set $\cal{S}$ containing pairs of integers $(A, B)$ such that for each $(A,B)\in \cal{S}$ we have $\gcd(A,B)$ is square-free and the Diophantine equation $x^2=Ak^n+B$ has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form $x^2=Ak^n+B$ with $k>2$, each containing five solutions in non-negative integers. %For example the equation $y^2=130\cdot 3^{n}+5550606$ has exactly five solutions with $n=0, 6, 11, 15, 16$. We also find new examples of equations $x^2=A2^{n}+B$ having six solutions in positive integers, e.g. the following Diophantine equations has exactly six solutions: \begin{equation*} \begin{array}{ll} x^2= 57\cdot 2^{n}+117440512 & n=0, 14, 16, 20, 24, 25, x^2= 165\cdot 2^{n}+26404 & n=0, 5, 7, 8, 10, 12. \end{array} \end{equation*} Moreover, based on an extensive numerical calculations we state several conjectures on the number of solutions of certain parametric families of the Diophantine equations of Ramanujan-Nagell type.