We give polynomialtime quantum algorithms for three problems from computational algebraic number theory. Motivation let d2z be a nonsquare positive integer. The solutions to pells equation have long been of interest to mathematicians, not least because of their value as approximations for n \sqrtn n. The nextin terms of magnitudesolution x1, y1 of pells equation can be found by expanding into. The first significant progress in solving the pells equation was made in india as early as a. Solving the pell equation cms books in mathematics. Many number theory problems can be transformed into the problem of pell equations solvability. A pell equation is a type of diophantine equation in the form for a natural number.
He posed the challenge to find a perfect square that, when multiplied by 92 and increased by 1, yields another perfect square. The study on solutions of the polynomial pell equation f 2. Dy 2 1 has infinitely many solutions in positive integers for a given d not being a perfect square, and that based on the least fundamental solution of the equation when we can simply calculate algebraically each remaining solution. Pells equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in algebraic number fields. Pells equation an equation of the form x2 dy2 1, where d is a positive integer that is not a perfect square and the equation is to be solved in integers. Mignotteon the number of solutions of simultaneous pell equations. Pell numbers arise historically and most notably in the rational approximation to v 2. Polynomialtime quantum algorithms for pells equation and. Publication date august 2000 topics mathematics, lecture digitizing sponsor msri language english. Next, for the numbers x0, y0, z0, t0 of cows of the same respective colors, the poet requires in lines 1726. Lagrange showed that every solution can easily be constructed from a so given d it suffices to compute a. We consider a slighty more general variant of the equation. Perform a suitable nonempty sequence of left and right steps, beginning with the pell quadratic form qd 1,0.
Primitive pythagorean triples and the negative pell equation. We begin by noting that for any positive integer d, the number p d is either an integer. If we apply conjugation to this equation, then using the fact that it is a. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Pells equation is a very simple, yet fundamental diophantine equation which is believed to have been known to mathematicians for over 2000 years. Pell s equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in algebraic number fields. Brahmagupta described how to use the known solution to a pells equation to create. Pells equation alternatively called the p ell f ermat equation is a type of a diophantine equation of the form x 2. How to find a fundamental solution to pells equation.
We have seen that the pell equation x2 dy2 1 is closely tied up with the problem of nding units in the ring of integers of the number eld k qp d, modulo the minor problem that zp d may just be an order in o k. Pells equation also called the pellfermat equation is any diophantine equation of the form. In general, the diophantine equation is the equation given by. He also uses this to imply that once we have have found one f. Following stiglers law of eponomy1, pells equation was studied by the indian mathematician and astronomer brahmagupta in 628 who discovered the composition law. Victor ufnarovski department of mathematics lth 1 abstract this is an initial study of pells equations of higher degree, which is an open problem in number theory. Primitive pythagorean triples and the negative pell equation keith matthews abstract abstract.
It is well known that the positive pell equation, x. Each step of the calculation will be displayed below the final solution. Pells equation is any diophantine equation of the form. Henceforth n will denote a positive integer that is not a square. However, the previous methods in determining the pell. Note pells equation always has the trivial solutions 1. Because of its popularity, the pell equation is often discussed in textbooks and recreational books concerning elementary. The history of this equation is long and circuitous, and involved a number of different approaches before a definitive theory was found. This paper uses continued fractions to give more explicit versions of results of a. Generally, is taken to be squarefree, since otherwise we can absorb the largest square factor into by setting. Cubic pells equation tien truong email protected johannes hedberggymnasiet march 15, 2012project course, 100p supervisor. We refer to 1 as a diophantine equation since only integer solutions are allowed. Pells equation is an important topic of algebraic number theory that involves quadratic forms and the structure of rings of integers in. To add icing, we can give explicit formulas for the solutions.
Pells equation article about pells equation by the. Pell equation is an important research object in elementary number theory of indefinite equation. As mentioned in chapter 2, the cattle problem reduces to that of solving the pell equation t 2. The smallest integer solution for x will be found, using bhaskara iis method. We shall always assume that d is positive but not a square, since otherwise there are. A note on the negative pell equation 1 introduction citeseerx. Our discussion of this topic follows the exposition of chapter vii of part one of elementary number theory by edmund landau. Williams relating the solvability of the negative pell equation to the existence of primitive. Solving the pell equation michael jacobson springer. Pells equation pells equation is the diophantine equation 1 x2 dy2 1 where d is a xed nonsquare positive integer. C i2z, and that xhas a purely periodic expansion if and. Gauss and lagrange made an equivalent but better method with reduced quadratic forms, that requires no decimal accuracy for the square root, just the integer part, just integer arithmetic, and no cycle. If n2n is a square, show the only solutions of x2 ny2 1 are 1. Barbeauspringer to my grandchildren alexander joseph gargaro maxwell edward gargaro victo.
In this article we formalize several basic theorems that correspond to pells equation. We would have, from which we can use casework to quickly determine the solutions. Victor ufnarovski department of mathematics lth 1 abstract this is an initial study of pell s equations of higher degree, which is an open problem in number theory. On the number of solutions to systems of pell equations. Pells equation is a very simple diophantine equation that has been known to mathematicians for over 2000 years. There were partial patterns and quite effective methods of finding solutions, but a complete. On polynomials solutions of quadratic diophantine equations. Pells equation is part of a central area of algebraic number theory that treats. Introduction pells equation has a nontrivial solution. Note that trivially x 1 and y 0 always solves equation 1. For example, the equation is known the linear diophantine equation. Note that if is a perfect square, then this problem can be solved using difference of squares.
In this paper we consider the solutions of the generalized matrix pell equations x 2. Solving the pell equation 5 respectively, one reads in lines 816 the restrictions x d. Abstract this is an initial study of pells equations of higher degree, which is an open problem in number theory. In cartesian coordinates, the equation has the form of a hyperbola. Complete solutions of the simultaneous pell equations x2. Hence, from now on, we will assume nis not a square. Contents 1 the computational task of solving pells equation 2 1. Pells equation solving x 2 ny 2 1 enter a positive nonsquare integer for n into the box and click go. Given a positive nonsquare integer d, pells equation is x2. If two large integers x and y form a solution to the pell equation.
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