Algorithmic Number Theory MSRIPublications Volume44,2008 Solving the Pell equation HENDRIK W. LENSTRA, JR. ABSTRACT. Weillustrate recent developments in computational number the- ory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the tradi- tional continued fraction method for solving the equation can be signicantly accelerated. The most promising method depends ...
Cambridge University Press 978-0-521-80854-5 - Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography Edited by J. P. Buhler and P. Stevenhagen Excerpt More information Algorithmic Number Theory MSRIPublications Volume44,2008 Solving the Pell equation HENDRIK W. LENSTRA, JR. ABSTRACT. Weillustrate recent developments in computational number the- ory by studying their implications for solving the Pell equation. We shall see that, if the solutions to ...
2 Problems Leading to Pell’s Equation and Preliminary Investigations Therstchapterpresentedasituationthatledtopairsofintegers(x,y)thatsatised equationsoftheformx22y2 kforsomeconstantk.Oneofthereasonsforthe popularity of Pell’s equation as a topic for mathematical investigation is the fact that many natural questions that one might ask about integers lead to a quadratic equation in two variables, which in turn can be cast as a Pell’s equation. In this chapter we will present ...
M.J.,.J. Jacobson, University of Calgary, AB, Canada; H.C. Williams, University of Calgary, AB, Canada Solving the Pell Equation Pell’s Equation is a very simple Diophantine equation that has been known to math- ematicians for over 2000 years. Even today research involving this equation contin- ues to be very active, as can be seen by the publication of at least 150 articles ...
Polynomial-Time Quantum Algorithms for Pell’s Equation and the Principal Ideal Problem∗ Sean Hallgren† NECLaboratories America, Inc. 4 Independence Way Princeton, NJ 08540 hallgren@nec-labs.com May 9, 2006 Abstract We give polynomial-time quantum algorithms for three problems from computa- tional algebraic number theory. The rst is Pell’s equation. Given a positive non- 2 2 square integer d, Pell’s equation is x − dy ...
Euler and Lagrange on Pell’s Equation Joshua Aaron McGill ID: 260352168 December 3, 2010 Introduction The eld of number theory is notorious for yielding immensely dicult problems that are deceptively easy to state. One such problem, known as Pell’s Equation, was studied by some of the greatest mathematicians th 1 in history and was not fully solved until the 18 century. Here we ...
Solving the Pell Equation H. W. Lenstra Jr. Pell’s Equation found in Euler’s Algebra [4, Abschn. 2, Cap. 7]. The Pell equation is the equation Modern textbooks usually give a formulation in terms of continued fractions, which is also due to 2 2 x =dy +1, Euler (see for example [9, Chap. 7]). Euler, as well to be solved in positive integers x ...
CONTINUED FRACTIONS AND PELL’S EQUATION SEUNGHYUNYANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell’s equation. I would like to thank my mentor Avan for introducing and guiding me through this extremely interesting material. I would like to cite Steuding’s detailed but slightly awed book as the main ...
Quantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE4: Hallgren’s algorithm for solving Pell’s equation In this and the next lecture, we will explore a nal application of the quantum Fourier transform over abelian groups, namely an algorithm discovered by Hallgren for solving a quadratic diophantine equation known as Pell’s equation. This algorithm is interesting for at least ...
MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000–000 S 0025-5718(XX)0000-0 MIDPOINT CRITERIA FOR SOLVING PELL’S EQUATION USING THE NEAREST SQUARE CONTINUED FRACTION KEITH MATTHEWS,JOHNROBERTSON,JIMWHITE Abstract. We derive midpoint criteria for Pell’s equation x2 − Dy2 = ±1, using the nearest square continued fraction expansion of √D. The period of the expansion is on average 70% that of ...
2 Pell’s Equation 2.1 Square-triangular numbers and Convergents of Continued Fractions Square-triangular numbers are integers which are simultaneously: • Perfect squares: of the form n2 for some n ∈ N; • Triangular: of the form m k = 1m(m+1) for some m ∈ N. ∑ 2 k=1 For example, 36 is a square-triangular number: Tondallsuch,weneedtosolvetheDiophantineequation2n2 = m(m+1). Itcanbeseenthatthis is equivalent to solving ...
Problem Books in Mathematics Edited by P. Winkler Problem Books in Mathematics Series Editor: Peter Winkler Pell’s Equation by Edward J. Barbeau Polynomials by Edward J. Barbeau Problems in Geometry by Marcel Berger, Pierre Pansu, Jean-Pic Berry, and Xavier Saint-Raymond Problem Book for First Year Calculus by George W. Bluman Exercises in Probability by T. Cacoullos Probability Through Problems by Marek Capin´ski and ...
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