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207
Some old problems and new results about quadratic forms
 Notices Amer. Math. Soc
, 1997
"... It may be a challenging problem to describe the integer solutions to a polynomial equation in several variables. Which integers, for example, are represented by a quadratic polynomial? This ..."
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It may be a challenging problem to describe the integer solutions to a polynomial equation in several variables. Which integers, for example, are represented by a quadratic polynomial? This
VALUES OF LUCAS SEQUENCES MODULO PRIMES
 THE ROCKY MOUNTAIN JOURNAL OF MATHEMATICS 33(2003), NO.3, 11231145.
, 2003
"... Let p be an odd prime, and a, b be two integers. It is the purpose of the paper to determine the values of u (p±1)/2(a, b) (mod p), where {un(a, b)} is the Lucas sequence ..."
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Cited by 16 (15 self)
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Let p be an odd prime, and a, b be two integers. It is the purpose of the paper to determine the values of u (p±1)/2(a, b) (mod p), where {un(a, b)} is the Lucas sequence
ConstantWeight Gray Codes for Local Rank Modulation
, 2010
"... We consider the local rankmodulation scheme in which a sliding window going over a sequence of realvalued variables induces a sequence of permutations. Local rankmodulation is a generalization of the rankmodulation scheme, which has been recently suggested as a way of storing information in flas ..."
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Cited by 16 (11 self)
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We consider the local rankmodulation scheme in which a sliding window going over a sequence of realvalued variables induces a sequence of permutations. Local rankmodulation is a generalization of the rankmodulation scheme, which has been recently suggested as a way of storing information in flash memory. We study constantweight Gray codes for the local rankmodulation scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation. We provide necessary conditions for the existence of cyclic and cyclic optimal Gray codes. We then specifically study codes of weight 2 and upper bound their efficiency, thus proving that there are no such asymptoticallyoptimal cyclic codes. In contrast, we study codes of weight 3 and efficiently construct codes which are asymptoticallyoptimal. We conclude with a construction of codes with asymptoticallyoptimal rate and weight asymptotically half the length, thus having an asymptoticallyoptimal charge difference between adjacent cells.
A characterization of the Squares in a Fibonacci string
 THEORETICAL COMPUTER SCIENCE
"... A (finite) Fibonacci string F n is defined as follows: F 0 = b, F 1 = a; for every integer n 2, F n = F n\Gamma1 F n\Gamma2 . For n 1, the length of F n is denoted by f n = jF n j. The infinite Fibonacci string F is the string which contains every F n , n 1, as a prefix. Apart from their general ..."
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A (finite) Fibonacci string F n is defined as follows: F 0 = b, F 1 = a; for every integer n 2, F n = F n\Gamma1 F n\Gamma2 . For n 1, the length of F n is denoted by f n = jF n j. The infinite Fibonacci string F is the string which contains every F n , n 1, as a prefix. Apart from their general theoretical importance, Fibonacci strings are often cited as worst case examples for algorithms which compute all the repetitions or all the "Abelian squares" in a given string. In this paper we provide a characterization of all the squares in F , hence in every prefix F n ; this characterization naturally gives rise to a \Theta(f n ) algorithm which specifies all the squares of F n in an appropriate encoding. This encoding is made possible by the fact that the squares of F n occur consecutively, in "runs", the number of which is \Theta(f n ). By contrast, the known general algorithms for the computation of the repetitions in an arbitrary string require \Theta(f n log f n ) time (and pro...
LowDimensional Lattices IV: The Mass Formula
 Proc. Royal Soc. London, A
, 1988
"... The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, ..."
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Cited by 15 (1 self)
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The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to compute. In particular we give a simple and reliable way to evaluate the 2adic contribution. Our version, unlike earlier ones, is visibly invariant under scale changes and dualizing. We use the formula to check the enumeration of lattices of determinant d 25 given in the first paper in this series. We also give tables of the "standard mass", the Lseries S (n / m)m  s (m odd), and genera of lattices of determinant d 25. 1.
Arithmetic of Elliptic Curves and Diophantine Equations
"... Introduction and background In 1952, P. Denes, from Budapest 1 , conjectured that three nonzero distinct nth powers can not be in arithmetic progression when n > 2 [15], i.e. that the equation x n + y n = 2z n has no solution in integers x, y, z, n with x #= y, and n > 2. One can ..."
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Cited by 14 (1 self)
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Introduction and background In 1952, P. Denes, from Budapest 1 , conjectured that three nonzero distinct nth powers can not be in arithmetic progression when n > 2 [15], i.e. that the equation x n + y n = 2z n has no solution in integers x, y, z, n with x #= y, and n > 2. One cannot fail to notice that it is a variant of the FermatWiles theorem. We would like to present the ideas which led H. Darmon and the author to the solution of Denes' problem in [13]. Many of them are those (due to Y. Hellegouarch, G. Frey, J.P. Serre, B. Mazur, K. Ribet, A. Wiles, R. Taylor, ...) which led to the celebrated proof of Fermat's last theorem. Others originate in earlier work of Darmon (and Ribet). The proof of Fermat's last theorem
Transformations of U(n + 1) multiple basic hypergeometric series
 UMEMURA (EDS.), PHYSICS AND COMBINATORICS: PROCEEDINGS OF THE NAGOYA 1999 INTERNATIONAL WORKSHOP (NAGOYA
, 1999
"... The purpose of this paper is to survey some of the main results and techniques from the transformation theory of U(n + 1) multiple basic hypergeometric series associated to the root system An. Our approach to this theory employs partial fraction decompositions, qdifference equations, and suitable ..."
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The purpose of this paper is to survey some of the main results and techniques from the transformation theory of U(n + 1) multiple basic hypergeometric series associated to the root system An. Our approach to this theory employs partial fraction decompositions, qdifference equations, and suitable multidimensional matrix inversions. These series were strongly motivated by Biedenharn and Louck and coworkers mathematical physics research involving angular momentum theory and the unitary groups U(n + 1), or equivalently An. The foundation of our theory is the U(n + 1) multiple sum renement of the terminating classical qbinomial theorem. This result contains as special, limiting, or transformed cases both the Macdonald identities for An, and U(n + 1) multiple sum extensions of the classical qbinomial theorem, Ramanujan's 1 1 sum, the balanced 3 2 summation theorem, and Rogers ' classical terminating verywellpoised 6 5 summation
Generalized Lambert series identities
 Proc. London Math. Soc. 91
, 2005
"... In 1944, F. J. Dyson dened the rank of a partition to be the largest part minus the number of parts and oered several conjectures, including combinatorial interpretations of Ramanujan’s famous congruences pð5nþ 4Þ 0 (mod 5) and pð7nþ 5Þ 0 (mod 7), where pðnÞ denotes the number of partitions of n. ..."
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Cited by 12 (2 self)
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In 1944, F. J. Dyson dened the rank of a partition to be the largest part minus the number of parts and oered several conjectures, including combinatorial interpretations of Ramanujan’s famous congruences pð5nþ 4Þ 0 (mod 5) and pð7nþ 5Þ 0 (mod 7), where pðnÞ denotes the number of partitions of n. These
Binomial Coefficients, the Bracket Function, and Compositions with Relatively Prime Summands.” The Fibonacci Quarterly Vol
, 1964
"... Bratter on the occasion of his 70th birthday. It is known [5] that a n e c e s s a r y and sufficient condition for p to be prime is that for every natural number n (1) Q 5 [  ] (modp). where Txl denotes the g r e a t e s t integer less than or equal to x. Indeed this result is equivalent to the c ..."
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Cited by 11 (2 self)
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Bratter on the occasion of his 70th birthday. It is known [5] that a n e c e s s a r y and sufficient condition for p to be prime is that for every natural number n (1) Q 5 [  ] (modp). where Txl denotes the g r e a t e s t integer less than or equal to x. Indeed this result is equivalent to the congruence k k (1 x) = 1 x (mod p) as is evident from the generating functions (2)