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Ramanujan’s ternary quadratic form
 Invent. Math
, 1997
"... In [R], S. Ramanujan investigated the representation of integers by positive definite quadratic forms, and the ternary form (1) φ1(x, y, z): = x2 + y2 + 10z2 was of particular interest to him. This form is in a genus consisting of two classes, and ..."
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In [R], S. Ramanujan investigated the representation of integers by positive definite quadratic forms, and the ternary form (1) φ1(x, y, z): = x2 + y2 + 10z2 was of particular interest to him. This form is in a genus consisting of two classes, and
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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Cited by 15 (2 self)
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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
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
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 13 (12 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
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
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 9 (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)
On the representation of unity by binary cubic forms
 Trans. Amer. Math. Soc
"... Abstract. If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) =1 possesses at most ten solutions in integers x and y, nineifF has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z ..."
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Abstract. If F (x, y) is a binary cubic form with integer coefficients such that F (x, 1) has at least two distinct complex roots, then the equation F (x, y) =1 possesses at most ten solutions in integers x and y, nineifF has a nontrivial automorphism group. If, further, F (x, y) is reducible over Z[x, y], then this equation has at most 2 solutions, unless F (x, y) is equivalent under GL2(Z)action to either x(x 2 − xy − y 2)orx(x 2 − 2y 2). The proofs of these results rely upon the method of ThueSiegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations F (x, y) =1forF cubic and irreducible of positive discriminant DF ≤ 10 6. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form F (x, y) =m and to Mordell’s equation y 2 = x 3 + k, wherem and k are nonzero integers. 1.
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 9 (9 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.
The Klein quartic in number theory
, 1999
"... Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � ..."
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Abstract. We describe the Klein quartic � and highlight some of its remarkable properties that are of particular interest in number theory. These include extremal properties in characteristics 2, 3, and 7, the primes dividing the order of the automorphism group of �; an explicit identification of � with the modular curve X(7); and applications to the class number 1 problem and the case n = 7 of Fermat.