Erdős and the Integers

Ruzsa, Imre Z.

doi:10.1006/jnth.1999.2395

Cited by 11 publications

(6 citation statements)

References 161 publications

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“…Then there exists a prime number l such that x < l < 2x by a famous result of Chebyshev (cf. [17]). Hence, let us consider two cases.…”

Section: New Bounds Of the Tensor Rankmentioning

confidence: 93%

On the tensor rank of the multiplication in the finite fields

Ballet

2008

Journal of Number Theory

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First, we prove the existence of certain types of non-special divisors of degree g − 1 in the algebraic function fields of genus g defined over F q . Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields F q by using Shimura and modular curves defined over F q . From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields F q , notably in the case of F 2 . These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in:

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“…Then there exists a prime number l such that x < l < 2x by a famous result of Chebyshev (cf. [17]). Hence, let us consider two cases.…”

Section: New Bounds Of the Tensor Rankmentioning

confidence: 93%

On the tensor rank of the multiplication in the finite fields

Ballet

2008

Journal of Number Theory

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“…The following theorem was proven by Ruzsa [32] in 1999 and is not as good as results obtained earlier; however, it is of some interest as will be described in the sequel. The following theorem was proven by Salem and Spencer [35].…”

Section: ∃A ⊆ [N]mentioning

confidence: 67%

Finding large 3-free sets I: The small n case

Gasarch

Glenn

Kruskal

2008

Journal of Computer and System Sciences

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There has been much work on the following question: given n, how large can a subset of {1, . . . , n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch applications of large 3-free sets, present techniques to find large 3-free sets of {1, . . . , n} for n 250, and give empirical results obtained by coding up those techniques. In the sequel we survey the known techniques for finding large 3-free sets of {1, . . . , n} for large n, discuss variants of them, and give empirical results obtained by coding up those techniques and variants.

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“…For Erdős's remarks on the work with Kalmar see [16, pp. 58-59] and Rusza [31,Section 1]. We mention also that the internal structure of prime factors of the middle binomial coefficient 2n n has received detailed study, see Erdős et al [15] and Pomerance [30].…”

Section: Introductionmentioning

confidence: 99%

Partial Factorizations of Products of Binomial Coefficients

Lagarias

2020

Preprint

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Let Gn = n k=0 n k , the product of the elements of the n-th row of Pascal's triangle. This paper studies the partial factorizations of Gn given by the product G(n, x) of all prime factors p of Gn having p ≤ x, counted with multiplicity. It shows log G(n, αn) ∼ f G (α)n 2 as n → ∞ for a limit function f G (α) defined for 0 ≤ α ≤ 1. The main results are deduced from study of functions A(n, x), B(n, x), that encode statistics of the base p radix expansions of the integer n (and smaller integers), where the base p ranges over primes p ≤ x. Asymptotics of A(n, x) and B(n, x) are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.

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Erdős and the Integers

Cited by 11 publications

References 161 publications

On the tensor rank of the multiplication in the finite fields

On the tensor rank of the multiplication in the finite fields

Finding large 3-free sets I: The small n case

Partial Factorizations of Products of Binomial Coefficients

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