Results 1  10
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114
Almost All Primes Can be Quickly Certified
"... This paper presents a new probabilistic primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], SolovayStrassen [SSI, and Rabin [R] ..."
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Cited by 69 (4 self)
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This paper presents a new probabilistic primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], SolovayStrassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption. Thc test terminates in expected polynomial time on all but at most an exponentially vanishing fraction of the inputs of length k, for every k. This result implies: • There exist an infinite set of primes which can be recognized in expected polynomial time. • Large certified primes can be generated in expected polynomial time. Under a very plausible condition on the distribution of primes in "small" intervals, the proposed algorithm can be shown'to run in expected polynomial time on every input. This
On the practical solution of the Thue equation
 INSTITUTE OF MATHEMATICS, UNIVERSITY OF DEBRECEN
, 1989
"... This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker’s theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented. ..."
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Cited by 44 (13 self)
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This paper gives in detail a practical general method for the explicit determination of all solutions of any Thue equation. It uses a combination of Baker’s theory of linear forms in logarithms and recent computational diophantine approximation techniques. An elaborated example is presented.
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
, 1996
"... Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n ..."
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Cited by 35 (1 self)
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Dedicated to the memory of GianCarlo Rota who encouraged me to write this paper in the present style Abstract. In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n 2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi’s special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular Cfractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the ηfunction identities of Macdonald. Moreover, the powers 4n(n + 1), 2n 2 + n, 2n 2 − n that appear in Macdonald’s work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing
Fibonacci and Galois Representations of FeedbackWithCarry Shift Registers
 IEEE Trans. Inform. Theory
, 2002
"... A feedbackwithcarry shift register (FCSR) with "Fibonacci" architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm. Like the linear feedback shift register (LFSR), the FCSR provides a simple and predictable method for the fast generation of p ..."
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Cited by 20 (2 self)
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A feedbackwithcarry shift register (FCSR) with "Fibonacci" architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm. Like the linear feedback shift register (LFSR), the FCSR provides a simple and predictable method for the fast generation of pseudorandom sequences with good statistical properties and large periods. In this paper, we describe and analyze an alternative architecture for the FCSR which is similar to the "Galois" architecture for the LFSR. The Galois architecture is more efficient than the Fibonacci architecture because the feedback computations are performed in parallel. We also describe the output sequences generated by theFCSR, a slight modification of the (Fibonacci) FCSR architecture in which the feedback bit is delayed for clock cycles before being returned to the first cell of the shift register. We explain how these devices may be configured so as to generate sequences with large periods. We show that the FCSR also admits a more efficient "Galois" architecture.
Super Ballot Numbers
, 1992
"... this paper were found with the help of the Maple symbolic algebra programming language. ..."
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Cited by 19 (3 self)
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this paper were found with the help of the Maple symbolic algebra programming language.
The determination of Gauss sums
 Bull. Amer. Math. Soc. (New Series
, 1981
"... geometric series can show that P\ 2 e 2 ™ / p 0, where/? is any integer exceeding one. Suppose that we replace n by n k ..."
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Cited by 18 (0 self)
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geometric series can show that P\ 2 e 2 ™ / p 0, where/? is any integer exceeding one. Suppose that we replace n by n k
On the topological Hochschild homology of bu. I.
 AMER. J. MATH
, 1993
"... The purpose of this paper and its sequel is to determine the homotopy groups of the spectrum THH(l). Here p is an odd prime, l is the Adams summand of plocal connective Ktheory (see for example [25]) and THH is the topological Hochschild homology ..."
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Cited by 17 (0 self)
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The purpose of this paper and its sequel is to determine the homotopy groups of the spectrum THH(l). Here p is an odd prime, l is the Adams summand of plocal connective Ktheory (see for example [25]) and THH is the topological Hochschild homology
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
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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Cited by 13 (0 self)
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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...