Solving more exponential diophantine equations with more modular arithmetic date. 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. Solve integer equations, determine remainders of powers, and much more with the power of modular arithmetic. The modulo arithmetic lesson aims to teach students addition and multiplication modulo n. We subsequently utilize these methods to completely solve such equations for various choices of the parameters a, b and c. Written out in this way, this method of nding remianders seems longer than just using standard long division. Almost as many methods to solve diophantine equations as equations. Systematizations and discussions on remainder problems in 18thcentury germany. The standard technique for solving this type of equation is manipulating the equation until the form, a n product of several expressions, an\text product of several expressions, a n product. Aug 12, 2014 linear congruence unique solution ba bsc 1st year number theory trigonometry exercise 2.
Finding all solutions in positive integers for such an equation is not an easy task. In some sense, modular arithmetic is easier than integer artihmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. The use of craig interpolants has enabled the development of power. Olympiad number theory through challenging problems. This idea of classi cation of methods have been taken from 5 and 17. Furthermore, if the equation is solvable mod p, then usually hensels lemma allows you to lift the solution to a solution mod any power of p. Explicit methods for solving diophantine equations arizona winter. Solve a linear congruence using euclids algorithm youtube. Diophantine equation, equation involving only sums, products, and powers in which all the constants are integers and the only solutions of interest are integers.
For each of the equations above, simply use euclids theorem to arrive at the equality chain. Fortunately, there is a well known algorithm for solving equations of this type. Chapter 2 presents classical diophantine equations, including linear, pythagorean, higherdegree, and exponential equations, such as catalans. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality. Now that you know fermat s little theorem, solve problem 3 in a mildly simpler way than before. There are five chapters on divisibility, modular arithmetic, padic valuation, diophantine equations, and problem solving strategies. An equation having one or more solutions is called solvable. We present a quick example of two nonlinear diophantine equations and prove they have no natural number solutions. I fear that the negative result of hilberts 10th problem is likely to mean that approaches are always going to be a bit ad hoc.
Efficient craig interpolation for linear diophantine disequations and linear modular equations article pdf available in formal methods in system design 351. This thesis is about the study of diophantine equations involving binary recur rent sequences with. Efficient craig interpolation for linear diophantine dis equations and linear modular equations article pdf available in formal methods in system design 351. This innocent sounding question leads to a vast field of interconnections. Applications include errorcorrecting codes, symmetric and public key. Diophantus wrote a treatise and he called arithmetica which is the earliest known book on algebra. We give a survey of methods used to connect the study of ternary diophantine equations to modern techniques coming from the theory of modular forms. Again, we dont even care about yexcept that it helps us get x. However, this technique cannot be used to show that solutions to a. Pdf applying modular arithmetic to diophantine equations. Solving pells equation or any other diophantine equation through modular arithmetic.
Generalizations of the subject led to important 19thcentury attempts to prove fermats last theorem and the development of significant parts of modern algebra. Finding the solution or solutions to a diophantine equation is closely tied to modular arithmetic and number theory. Solving diophantine equations using modular arithemtic. You will get even more practice on a quiz when youve finished the lesson. Pdf efficient craig interpolation for linear diophantine. An introduction to diophantine equations a problembased. Modular arithmetic is a special type of arithmetic that involves only integers.
Begin higher degree diophantine equations in chapter 5. Modular arithmetic operations considering the remainder modulo an integer is a powerful, foundational tool in number theory. We can represent each integer modulo mby one of the standard representatives, just like we can write any fraction in a reduced form. Furthermore, if the equation is solvable mod p, then usually hensels lemma allows you to lift the solution to a solution mod any. This is an example of an exponential diophantine equation.
Prove the existence part of the division algorithm. For example, suppose we wish to solve 15x 33 pmod 69q. Concerning a diophantine equation three basic problems arise. A ring consists of a set r with elements 0,1 2 r, and binary. 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. Solving linear diophantine equations with congruences. Classical and modular approaches to exponential diophantine equations i. Zn satisfying 1 is called a solution to equation 1. Diophantine equations with powers this section deals with equations with terms of the form a n an a n, where a a a is a given positive integer. Modular arithmetic is important in number theory, where it is a fundamental tool in the solution of diophantine equations particularly those restricted to integer solutions.
Simple modular arithmetic considerations like parity help to. An equation with no integer solutions number theory an equation with no integer solutions. Now that you know fermat s little theorem, solve problem 3 in a. There are many other representatives which could be used, however, and this will be important in the next section. On the solutions of quadratic diophantine equations ii murata, manabu and yoshinaga, takashi, journal of the mathematical society of japan, 2018. Rings and modular arithmetic so far, we have been working with just one operation at a time.
Arithmetic of elliptic curves and 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 equations, etc. Systems of proportionally modular diophantine inequalities. Solving to find the general equation with a mo d equation. Introduction to number theory with applications to middle school mathematics and cryptography. Number theory modular arithmetic stanford university. Modular arithmetic in this chapter, well look at some applications of modular arithmetic, i. A diophantine equation is an algebraic equation for which rational or integral solutions are sought. Classical and modular approaches to exponential diophantine. In particular, well get applications to divisibility tests, necessary conditions for solutions of various diophantine equations including nonsolvability. Solving linear diophantine equations and linear congruential. If we solve one of these congruences, then we will obtain solutions for the linear diophantine equation corresponding to them. Since part 3 of the hypothesis holds, we may invoke part 3 of theorem.
Lastly, there is the case where the legendre symbol equality. Glossary of arithmetic and diophantine geometry wikipedia. Lecture on continued fractions pavel guerzhoy pdf continued fractions and modular forms, seminar by ilan vardi summary by. Our main results are polynomial time algorithms for obtaining interpolants for conjunctions of linear diophantine equations, linear modular equations linear congruences, and linear diophantine disequations. Pdf the set of integer solutions to the inequality axmodb. Diophantine equations, geometry of numbers, irrationality. The standard technique for solving this type of equation is manipulating the equation until the form, a n product of several expressions, an\text product of several expressions, a n product of several expressions, is obtained. Modular arithmetic and diophantine equations physics forums. The modular arithmetic contradiction trick for diophantine. Modular arithmetic victor adamchik fall of 2005 plan 1. Tools to deal with diophantine equations here i will describe the general tools one can use to approach a diophantine equation. Many such equations can be solved using modular arithmetic, often quite a bit of modular arithmetic.
On modular equations pierpont, james, bulletin of the american mathematical society, 1897. Named in honour of the 3rdcentury greek mathematician. My goal for the text was to show how several problem solving skillsexperimenting with small cases, simplification, and pattern findingcan be used to tackle a wide variety of olympiad number theory problems. This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic. The study of diophantine equations is the study of solutions of polynomial equations or systems. The work uniquely presents unconventional and nonroutine. This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of diophantine equations to encompass large parts of number theory and algebraic geometry. Diophantine equations and its applications in real life. Quadratic diophantine equations with applications to quartic equations choudhry, ajai, rocky mountain journal of mathematics, 2016.
Solving an exponential diophantine equation with modular. Furthermore, if the equation is solvable mod p, then usually hensel. The modular approach to diophantine equations mathematical. This is a first draft version of workbooktext for mtht 467. Himanshu jain 1, edmund clarke, and orna grumberg2 1 school of computer science, carnegie mellon university 2 department of computer science, technion israel institute of technology abstract. High school mathematics extensionsprimesmodular arithmetic. Diophantine equations modular arithmetic considerations. Using modular arithmetic, the algorithm transforms the diophantine equation to a linear recurrence relation. Lncs 5123 efficient craig interpolation for linear. How can one natural number be expressed as the product of smaller natural numbers.
We know that there are integers r and s for which adr and b ds. Sometimes, modular arithmetic can be used to prove that no solutions to a given diophantine equation exist. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and padic fields but not the real numbers which are used in real algebraic geometry. Diophantine equations with arithmetic functions and binary. Fibonacci and lucas perfect powers by yann bugeaud, maurice mignotte, and samir siksek abstract this is the. Olympiad number theory through challenging problems justin stevens. Since by dirichlets theorem on primes in arithmetic progressions there.
Denes, from budapest 1, conjectured that three nonzero distinct nth powers can not be in arithmetic progression when n 2 15, i. Hilberts 10th problem was basically asking for an algorithm for solving or showing unsolvability of diophantine equations. Solving more exponential diophantine equations with more. Oct 25, 2015 mix play all mix maths with jay youtube intro to chinese remainder theorem and eulers totient theorem via a challenging problem duration. Introduction to number theory with applications to middle. In fact, finding the inverse of a number in modular arithmetic amounts to solving a type of equations called diophantine equations. With applications to middle school mathematics and cryptography supplementary materials. If none of them work then we know \y\ does not have an inverse. Linear diophantine equations got their name from diophantus. Diophantine equations and congruences 297 there exists z. Mix play all mix maths with jay youtube intro to chinese remainder theorem and eulers totient theorem via a challenging problem duration. What you have is an exponential diophantine equation.
As an example, we should try to find the inverse of 216 in mod 811. A few nonlinear diophantine equations without solutions. Solve the following diophantine equations evaluate or solve, and show your methods. Specifically, if we show that the equation in question is never true mod, for some integer, then we have shown that the equation is false. Diophantus of alexandria was a mathematician who lived around the 3rd century. Silverman, advanced topics in the arithmetic of elliptic curves, springerverlag, gtm. The division algorithm also works in qx, the set of polynomials with rational coe cients, and rx, the set of all polynomials with real coe cients. Firstly, and for this purpose, the empirical algorithm for obtaining experimental results is presented for diophantine equations of polynomialtype.