Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients

Granville, Andrew; Ramaré, Olivier

doi:10.1112/s0025579300011608

Cited by 64 publications

(40 citation statements)

References 12 publications

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“…Erdős made a number of conjectures quantifying these observations, one well-known conjecture being that the middle coefficient 2n n is not squarefree for any n > 4. Granville and Ramaré [190] proved this conjecture in 1996. The corresponding problem for q-binomial and, more generally, q-multinomial coefficients is much more complicated.…”

Section: )mentioning

confidence: 82%

Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on 'elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on 'Artin for K-theory of number fields', and Hester Graves (together with me) on 'Artin's conjecture and Euclidean domains'.

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Section: )mentioning

confidence: 82%

Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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“…Granville and Ramaré have verified this theorem for 2082 ≤ n ≤ 10 10 by using a direct consequence of Kummer's theorem, and for n ≥ 2 1617 by using bounds on exponential sums [2]. By using the following proposition of Granville and Ramaré, it becomes a practical computational problem to establish this theorem for 10 10 ≤ n ≤ 2 1617 .…”

Section: Other Calculationsmentioning

confidence: 88%

Finding prime pairs with particular gaps

Cutter

2001

Math. Comp.

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“…The best lower bound was recently established by Granville and Ramaré [5] who proved that there exists an absolute positive constant c such that g(k) > exp(c(log 3 k/ log log k)…”

Section: Introductionmentioning

confidence: 99%

Further tabulation of the Erdös-Selfridge function

Lukes¹,

Scheidler²,

Williams³

1997

Math. Comp.

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Abstract. For a positive integer k, the Erdös-Selfridge function is the least integer g(k) > k + 1 such that all prime factors of g (k) k exceed k. This paper describes a rapid method of tabulating g(k) using VLSI based sieving hardware. We investigate the number of admissible residues for each modulus in the underlying sieving problem and relate this number to the size of g(k). A table of values of g(k) for 135 ≤ k ≤ 200 is provided.

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Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients

Cited by 64 publications

References 12 publications

Artin's Primitive Root Conjecture – A Survey

Artin's Primitive Root Conjecture – A Survey

Finding prime pairs with particular gaps

Further tabulation of the Erdös-Selfridge function

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