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PolynomialTime Quantum Algorithms for Pell's Equation and the Principal Ideal Problem
 in Proceedings of the 34th ACM Symposium on Theory of Computing
, 2001
"... Besides Shor's polynomialtime quantum algorithms for factoring and discrete log, all progress in understanding when quantum algorithms have an exponential advantage over classical algorithms has been through oracle problems. Here we give efficient quantum algorithms for two more nonoracle problems ..."
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Cited by 82 (7 self)
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Besides Shor's polynomialtime quantum algorithms for factoring and discrete log, all progress in understanding when quantum algorithms have an exponential advantage over classical algorithms has been through oracle problems. Here we give efficient quantum algorithms for two more nonoracle problems. The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation is x²  dy² = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem is the principal ideal problem in real quadratic number fields. Solving this problem is at least as hard as solving Pell's equation, and is the basis of a cryptosystem which is broken by our algorithm. We also state some related open problems from the area of computational algebraic number theory.
Notes on Hallgren’s efficient quantum algorithm for solving Pell’s equation
, 302
"... Pell’s equation is x 2 − dy 2 = 1 where d is a squarefree integer and we seek positive integer √ solutions x,y> 0. Let (x0, y0) be the smallest solution (i.e. having smallest A = x0 + y0 d). Lagrange showed that every solution can easily be constructed from A so given d it suffices to compute A. It ..."
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Cited by 6 (0 self)
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Pell’s equation is x 2 − dy 2 = 1 where d is a squarefree integer and we seek positive integer √ solutions x,y> 0. Let (x0, y0) be the smallest solution (i.e. having smallest A = x0 + y0 d). Lagrange showed that every solution can easily be constructed from A so given d it suffices to compute A. It is known that A can be exponentially large in d so just to write down A we need exponential time in the input size log d. Hence we introduce the regulator R = lnA and ask for the value of R to n decimal places. The best known classical algorithm has subexponential running time O(exp √ log d,poly(n)). Hallgren’s quantum algorithm gives the result in polynomial time O(poly(log d),poly(n)) with probability 1/poly(log d). The idea of the algorithm falls into two parts: using the formalism of algebraic number theory we convert the problem of solving Pell’s equation into the problem of determining R as the period of a function on the real numbers. Then we generalise the quantum Fourier transform period finding algorithm to work in this situation of an irrational period on the (not finitely generated) abelian group of real numbers. These notes are intended to be accessible to a reader having no prior acquaintance with algebraic number theory; we give a self contained account of all the necessary concepts and we give elementary proofs of all the results needed. Then we go on to describe Hallgren’s generalisation of the quantum period finding algorithm, which provides the efficient computational solution of Pell’s equation in the above sense. CONTENTS 1 The computational task of solving Pell’s equation 2 1.1 Approach for the efficient quantum algorithm 4 1.2 Note to the reader 4 2 Algebraic
Higher Descent on Pell Conics. I. From Legendre to Selmer, preprint 2003; cf. p
"... The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952], ..."
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Cited by 5 (4 self)
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The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952],
Heegner points, Stark–Heegner points, and values
 of Lseries, inProceedings of the ICM
, 2006
"... Abstract. Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch an ..."
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Cited by 3 (2 self)
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Abstract. Elliptic curves over Q are equipped with a systematic collection of Heegner points arising from the theory of complex multiplication and defined over abelian extensions of imaginary quadratic fields. These points are the key to the most decisive progress in the last decades on the Birch and SwinnertonDyer conjecture: an essentially complete proof for elliptic curves over Q of analytic rank ≤ 1, arising from the work of GrossZagier and Kolyvagin. In [Da2], it is suggested that Heegner points admit a host of conjectural generalisations, referred to as StarkHeegner points because they occupy relative to their classical counterparts a position somewhat analogous to Stark units relative to elliptic or circular units. A better understanding of StarkHeegner points would lead to progress on two related arithmetic questions: the explicit construction of global points on elliptic curves (a key issue arising in the Birch and SwinnertonDyer conjecture) and the analytic construction of class fields sought for in Kronecker’s Jugendtraum and Hilbert’s twelfth problem. The goal of this article is to survey Heegner points, StarkHeegner points, their arithmetic applications and their relations (both proved, and conjectured) with special values of Lseries attached to modular forms.
COMPLEXITY RESULTS FOR CR MAPPINGS BETWEEN SPHERES
, 708
"... Abstract. Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here th ..."
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Cited by 2 (1 self)
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Abstract. Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation. We then sharpen our results and obtain various congruences guaranteeing nonuniqueness. We also show that a gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem.
World Academy of Science, Engineering and Technology 43 2008 The Pell Equation x 2 − (k 2 − k)y 2 =2 t
"... Abstract—Let k, t, d be arbitrary integers with k ≥ 2, t ≥ 0 and d = k 2 − k. In the first section we give some preliminaries from Pell equations x 2 − dy 2 =1and x 2 − dy 2 = N, where N be any fixed positive integer. In the second section, we consider the integer solutions of Pell equations x 2 − d ..."
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Abstract—Let k, t, d be arbitrary integers with k ≥ 2, t ≥ 0 and d = k 2 − k. In the first section we give some preliminaries from Pell equations x 2 − dy 2 =1and x 2 − dy 2 = N, where N be any fixed positive integer. In the second section, we consider the integer solutions of Pell equations x 2 − dy 2 =1and x 2 − dy 2 =2 t.We give a method for the solutions of these equations. Further we derive recurrence relations on the solutions of these equations. Keywords—Pell equation, solutions of Pell equation. I. PRELIMINARY FACTS. Let d ̸ = 1be a positive nonsquare integer and N be any fixed positive integer. Then the equation x 2 − dy 2 = ±N (1) is known as Pell equation and is named after John Pell (16111685), a mathematician who searched for integer solutions to equations of this type in the seventeenth century. Ironically, Pell was not the first to work on this problem, nor did he contribute to our knowledge for solving it. Euler (17071783), who brought us the ψfunction, accidentally named the equation after Pell, and the name stuck. For N =1, the Pell equation x 2 − dy 2 = ±1 (2) is known as the classical Pell equation and was first studied by Brahmagupta (598670) and Bhaskara (11141185), (see [1]). Its complete theory was worked out by Lagrange (17361813), not Pell. It is often said that Euler (17071783) mistakenly attributed Brouncker’s (16201684) work on this equation to Pell. However the equation appears in a book by Rahn (16221676) which was certainly written with Pell’s help: some say entirely written by Pell. Perhaps Euler knew what he was doing in naming the equation. Baltus [2], Kaplan and
3.3 Linear forms
, 2010
"... 3.3.1 Siegel’s method: m +1 linear forms For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simult ..."
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3.3.1 Siegel’s method: m +1 linear forms For proving linear independence of real numbers, Hermite [18] considered simultaneous approximation to these numbers by algebraic numbers. The point of view introduced by Siegel in 1929 [34] is dual (duality in the sense of convex bodies): he considers simultaneous approximation by means of independent linear forms. We define the height of a linear form L = a0X0 + · · · + amXm with complex coefficients by H(L) = max{a0,..., am}. Lemma 17. Let ϑ1,..., ϑm be complex numbers. Assume that, for any ɛ> 0, there exists m +1 linearly independent linear forms L0,..., Lm in m +1 variables, with coefficients in Z, such that max 0≤k≤m Lk(1, ϑ1,..., ϑm)  < ɛ where H = max
1.1 Early history
"... 1.1.1 Simple proofs of irrationality The early history of irrationality goes back to the Greek mathematicians Hippasus of Metapontum (around 500 BC) and Theodorus of Cyrene, Eudoxus, Euclid. There are different early references in the Indian civilisation and the ..."
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1.1.1 Simple proofs of irrationality The early history of irrationality goes back to the Greek mathematicians Hippasus of Metapontum (around 500 BC) and Theodorus of Cyrene, Eudoxus, Euclid. There are different early references in the Indian civilisation and the