Jesmanowicz conjecture on exponential diophantine equations miyazaki, takafumi, functiones et approximatio commentarii mathematici, 2011. Today, we will begin by focusing on a special kind of diophantine. Diophantine equations 1 exponential diophantine equations. These properties will then provide necessary and su. Examples include the ramanujannagell equation, 2 n. Abstract this paper is focused on solutions of two diophantine equations of the type. Browse other questions tagged diophantine equations or ask your own question. This suggests that the modular approach should be applied to exponential diophantine equations. Hilberts 10th problem 14 asked if an algorithm exists for determining whether an arbitrary diophantine equation has a solution or not. Topics covered include the thue equations, the generalised hyperelliptic equation, and the fermat and catalan equations. On some ternary pure exponential diophantine equations.
On some special diophantine equations rosenthall, e. Pdf small twovariable exponential diophantine equations. On a conjecture on exponential diophantine equations 3 the last part of the paper is devoted to improvements of bounds on the parameters associated to a putative solution to system 2. In this paper, a classification of all positive integer solutions of the equation is given. Further, by an elementary approach, we prove that if, then the equation has only the positive integer solution, except for and, where is a positive integer with. Equations with integer coefficients to which integer solutions are sought.
On the exponential diophantine equations of degree three. If a diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential diophantine equation. There exists no algorithm for the determination of solvability of arbitrary exponential diophantine equations. In this paper, by an argument similar to that used in theorems a, b, we prove the following. Pdf on a conjecture on exponential diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An introduction to diophantine equations springerlink. Pdf to text batch convert multiple files software please purchase personal license. Find out information about exponential diophantine equation. Applying the same argument as before, we may see that. On a conjecture on exponential diophantine equations.
In this paper, using some properties of exponential diophantine equations. Solving linear diophantine equations and linear congruential. Tijdeman, exponential diophantine equations, cambridge univ. In this paper, we consider the diophantine equation. Trinomials and exponential diophantine equations universiteit. A diophantine equation is an equation relating integer or sometimes natural number or whole number quanitites finding the solution or solutions to a diophantine equation is closely tied to modular arithmetic and number theory.
Classical and modular approaches to exponential diophantine equations. Linear diophantine equations got their name from diophantus. As is well known, many types of diophantine problems are associated to linear forms in logarithms of algebraic numbers see, e. The brute force approach for solving diophantine equation is a well. Exponential diophantine equation definition of exponential. On some ternary pure exponential diophantine equations with. Pdf the exponential diophantine equation researchgate. If n 2, there are no nontrivial1 solutions in integers to the equation. Optical solutions for linear diophantine equations. Chapter 2 presents classical diophantine equations, including linear, pythagorean, higherdegree, and exponential equations, such as catalans. Maxmin of polynomials and exponential diophantine equations.
Counting solutions to diophantine equations oscar marmon 1 introduction the study of diophantine equations is among the oldest branches of mathematics, and also one of the most intriguing. In order to solve this equation, we will consider two cases depend on the parity of. Diophantus of alexandria was a mathematician who lived around the 3rd century. However, it is known that every exponential diophantine equation could be transformed mechanically into an equivalent ordinary diophantine equation in more. Often, when a diophantine equation has infinitely many solutions, parametric form is used to express the relation between the variables of the equation. Exponential diophantine equations involving products of. The authors present, in a clear and unified fashion, applications to exponential diophantine equations and linear recurrence sequences of the gelfondbaker theory of linear forms in logarithms of algebraic numbers. They are meant to shrink the search domains for the components of a solution to a manageable size according to the presentday technology. Exponential diophantine equation synonyms, exponential diophantine equation pronunciation, exponential diophantine equation translation, english dictionary definition of exponential diophantine equation.
Because the results are restricted to integers, different algorithms must be used. Raghavan, tata institute of fundamental research, bombay 1988, 217229. From this it follows that there does not exist an algorithm to decide solvability of diophantine equations. Solving exponential diophantine equations using lattice basis. Let lbe a prime with l 3 mod 8, and let a, b, cbe xed positive.
This problemsolving book is an introduction to the study of diophantine equations, a class of equations in which only integer solutions are allowed. Exponential diophantine equation article about exponential. If a diophantine equation has variables as exponents, it is said to be exponential diophantine equation. In this lecture, we will introduce some basic questions and conjectures and explain what thue proved. Press 1987 mr1766003 mr1764814 mr35008 mr14434 mr0972011 mr0891406 zbl 1156. On some ternary pure exponential diophantine equations with three consecutive positive integers bases ruiqin fu1,bohe2, hai yang3 and huilin zhu4. More generally, ifkc is a field andac a ring, we will. Exponential diophantine equations mathematical sciences.
Upper bounds for solutions of an exponential diophantine equation miyazaki, takafumi, rocky mountain journal of mathematics, 2015. Finding integral solutions is more difficult than a standard solution and requires an ordered pattern of steps. Diophantus wrote a treatise and he called arithmetica which is the earliest known book on algebra. The undecidability of exponential diophantine equations julia robinson university of california, berkeley, california, u. It is the challenge to provide a general algorithm which, for any given diophantine equation a polynomial equation with integer coefficients and a finite number of unknowns, can decide whether the equation has a solution with all unknowns taking integer values. In a certain sense we extend the results of c2q,1,1 in 6 to exponential polynomials.
Diophantine analysis deals with various techniques of solving diophantine equations. This is the first in a series of papers whereby we combine the classical approach to exponential diophantine equations linear forms in logarithms, thue. Then equation 2 has only the positive integer solution x. Exponential diophantine equations and inequalities let s q be the ring ofpower sums, i. A search for similar results in online resources like the online encyclopedia of integer sequences reveals the lack of a concentrated effort in this direction. Small twovariable exponential diophantine equations 8 views the equation modulo 39, one gets y 5 mod 38. In this paper, using some properties of exponential diophantine equations and some results on the existence of. Matijasevic proved 11 that the exponential relation y 2 x is diophantine this together with 2 implies that every recursively enumerable set is diophantine, i. You must first find the greatest common factor of the coefficients in the problem, and then use that result.
Let and be fixed coprime odd positive integers with. When we calculate x, we see that x 0 mod 9 modulo any of the primes 19, 37, or 109. Using certain results of exponential diophantine equations, we prove that i if, then the equation has no positive integer solutions. Fibonacci and lucas perfect powers yann bugeaud, maurice mignotte, samir siksek abstract. Identical to proof for diophantine equations from the full mrdp theorem. Probably the most famous diophantine equation is the one representing fermats last theorem. A problembased approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants including olympiad and putnam competitors as well as readers interested in essential mathematics. An introduction to diophantine equations a problembased. A diophantine equation is an algebraic equation for which rational or integral solutions are sought. The work uniquely presents unconventional and nonroutine. Oct 04, 2015 1 exponential diophantine equations diophantine equations are just equations we solve with the constraint that all variables must be integers.
This is the second in a series of papers where we combine the classical approach to exponential diophantine equations linear forms in logarithms, thue equations, etc. These are generally really hard to solve for example, the famous fermats last theorem is an example of a diophantine equation. Hilberts tenth problem asks for an algorithm to determine whether or not an arbitrary diophantine equation has a solution in integers. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
A search for similar results in online resources like the online encyclopedia. Diophantine geometrywhich is the application of techniques from algebraic geometry in this field, has continued to grow as a result. There are many, many forms of diophantine equations, but equations of the sort that. Pdf a note on the ternary purely exponential diophantine.
We prove that the distribution of their singular values converges to the local marchenkopastur law at scales nd for an. However, formatting rules can vary widely between applications and fields of interest or study. Shorey, some exponential diophantine equations ii, number theory and related topics, ed. Universal diophantine equation the journal of symbolic. How to solve a linear diophantine equation with pictures. Many questions in number theory concern perfect powers, numbers of the form ab where a and b are rational integers with 1, 61. The undecidability of exponential diophantine equations. The presentation features some classical diophantine equations, including linear, pythagorean, and some higher degree equations, as well as exponential diophantine equations. Hilberts tenth problem is the tenth on the list of mathematical problems that the german mathematician david hilbert posed in 1900. Exponential diophantine equations and inequalities. Using certain results of exponential diophantine equations, we prove that i if p. Exponential diophantine equations semantic scholar. Solving a linear diophantine equation means that you need to find solutions for the variables x and y that are integers only. Exponential diophantine equation mathematics stack exchange.
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