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Approximate integer common divisors
 CaLC 2001, LNCS
, 2001
"... Abstract. We show that recent results of Coppersmith, Boneh, Durfee and HowgraveGraham actually apply in the more general setting of (partially) approximate common divisors. This leads us to consider the question of “fully ” approximate common divisors, i.e. where both integers are only known by ap ..."
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Cited by 40 (1 self)
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Abstract. We show that recent results of Coppersmith, Boneh, Durfee and HowgraveGraham actually apply in the more general setting of (partially) approximate common divisors. This leads us to consider the question of “fully ” approximate common divisors, i.e. where both integers are only known
Approximate common divisors via lattices
, 2012
"... We analyze the multivariate generalization of HowgraveGraham’s algorithm for the approximate common divisor problem. In the mvariable case with modulus N and approximate common divisor of size N β, this improves the size of the error tolerated from N β2 to N β(m+1)/m, under a commonly used heuri ..."
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Cited by 23 (1 self)
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We analyze the multivariate generalization of HowgraveGraham’s algorithm for the approximate common divisor problem. In the mvariable case with modulus N and approximate common divisor of size N β, this improves the size of the error tolerated from N β2 to N β(m+1)/m, under a commonly used
Multidimensional greatest common divisor and Lehmer algorithms
 BIT
, 1977
"... Abstract. A class of multidimensional greatest common divisor algorithms is studied. Their connection with the Jacobi algorithm is established and used to obtain theoretical properties such as the existence of digit frequencies. A technique of D. H. Lehmer's for Euclid's algorithm is gener ..."
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Cited by 1 (0 self)
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Abstract. A class of multidimensional greatest common divisor algorithms is studied. Their connection with the Jacobi algorithm is established and used to obtain theoretical properties such as the existence of digit frequencies. A technique of D. H. Lehmer's for Euclid's algorithm
Integral Cayley Graphs Defined by Greatest Common Divisors
 ELECTRON. J. COMBIN
, 2011
"... An undirected graph is called integral, if all of its eigenvalues are integers. Let Γ = Zm1 ⊗ · · · ⊗Zmr be an abelian group represented as the direct product of cyclic groups Zmi of order mi such that all greatest common divisors gcd(mi,mj) ≤ 2 for i = j. We prove that a Cayley graph Cay(Γ,S) ..."
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Cited by 3 (1 self)
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An undirected graph is called integral, if all of its eigenvalues are integers. Let Γ = Zm1 ⊗ · · · ⊗Zmr be an abelian group represented as the direct product of cyclic groups Zmi of order mi such that all greatest common divisors gcd(mi,mj) ≤ 2 for i = j. We prove that a Cayley graph Cay
LIMIT THEOREMS FOR EMPIRICAL DENSITY OF GREATEST COMMON DIVISORS
"... Abstract. The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical density. We will also obtain a sharp rate of convergen ..."
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Abstract. The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical density. We will also obtain a sharp rate
On the Fourier transform of the greatest common divisor, Integers 13
, 2013
"... The discrete Fourier transform of the greatest common divisor îd[a](m) = m∑ k=1 gcd(k,m)αkam, with αm a primitive mth root of unity, is a multiplicative function that generalises both the gcdsum function and Euler’s totient function. On the one hand it is the Dirichlet convolution of the identity ..."
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Cited by 1 (0 self)
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The discrete Fourier transform of the greatest common divisor îd[a](m) = m∑ k=1 gcd(k,m)αkam, with αm a primitive mth root of unity, is a multiplicative function that generalises both the gcdsum function and Euler’s totient function. On the one hand it is the Dirichlet convolution
On primitive recursive algorithms and the greatest common divisor function
 Theor. Comput. Sci
, 2003
"... Abstract. We establish linear lower bounds for the complexity of nontrivial, primitive recursive algorithms from piecewise linear given functions. The main corollary is that logtime algorithms for the greatest common divisor from such givens (such as Stein’s) cannot be matched in efficiency by prim ..."
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Cited by 4 (2 self)
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Abstract. We establish linear lower bounds for the complexity of nontrivial, primitive recursive algorithms from piecewise linear given functions. The main corollary is that logtime algorithms for the greatest common divisor from such givens (such as Stein’s) cannot be matched in efficiency
ADDENDA TO “ON COMMON DIVISORS OF MULTINOMIAL COEFFICIENTS”
, 2010
"... Abstract. For k and N positive integers, let us understand a “proper knomial coefficient of weight N ” to mean an integer (n1 + · · · +nk)!/(n1!... nk!) where all ni are positive integers, and n1 + · · · +nk = N. Erdős and Szekeres [3] show that any two proper binomial coefficients of equal we ..."
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Cited by 1 (1 self)
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weight have a common divisor> 1. The analogous statement for k knomial coefficients (k> 1) was conjectured in 1997 by David Wasserman. In [1], after proving a conjecture of Erdős and Szekeres on the growth of the g.c.d. of two binomial coefficients of equal weight, I obtained some restrictions
The Greatest Common Divisor of Two Recursive Functions
"... Let g, h be solutions of a linear recurrence relation of length 2. We show that under some mild assumptions the greatest common divisor of g(n) and h(n) is periodic as a function of n and compute its mean value. 1. Problems and Results Let a, b be coprime integers, b ̸ = 0, and consider the recurren ..."
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Let g, h be solutions of a linear recurrence relation of length 2. We show that under some mild assumptions the greatest common divisor of g(n) and h(n) is periodic as a function of n and compute its mean value. 1. Problems and Results Let a, b be coprime integers, b ̸ = 0, and consider
Results 1  10
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36,244