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1. CJM 2009 (vol 62 pp. 582)
On the Distribution of Pseudopowers An x-pseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.
Categories:11A07, 11L07, 11N36 |
2. CJM 2009 (vol 61 pp. 481)
Uniform Distribution of Fractional Parts Related to Pseudoprimes We estimate exponential sums with the Fermat-like quotients
$$
f_g(n) = \frac{g^{n-1} - 1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n-1}-1}{P(n)},
$$
where $g$ and $n$ are positive integers, $n$ is composite, and
$P(n)$ is the largest prime factor of $n$. Clearly, both $f_g(n)$
and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base
$g$, and if $n$ is a Carmichael number, this is true for all $g$
coprime to $n$. Nevertheless, our bounds imply that the fractional
parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on
average over~$g$ for $f_g(n)$, and individually for $h_g(n)$. We
also obtain similar results with the functions ${\widetilde f}_g(n)
= gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.
Categories:11L07, 11N37, 11N60 |
3. CJM 2009 (vol 61 pp. 336)
The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes |
The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes Let $\lambda$ be a fixed integer exceeding $1$ and $s_n$ any
strictly increasing sequence of positive integers satisfying $s_n\le
n^{15/14+o(1)}.$ In this paper we give a version of the large sieve
inequality for the sequence $\lambda^{s_n}.$ In particular, we
obtain nontrivial estimates of the associated trigonometric sums
``on average" and establish equidistribution properties of the
numbers $\lambda^{s_n} , n\le p(\log p)^{2+\varepsilon}$,
modulo $p$ for most primes $p.$
Keywords:Large sieve, exponential sums Categories:11L07, 11N36 |
4. CJM 2005 (vol 57 pp. 338)
Certain Exponential Sums and Random Walks on Elliptic Curves For a given elliptic curve $\E$, we obtain an upper bound
on the discrepancy of sets of
multiples $z_sG$ where $z_s$ runs through a sequence
$\cZ=\(z_1, \dots, z_T\)$
such that $k z_1,\dots, kz_T $ is a permutation of
$z_1, \dots, z_T$, both sequences taken modulo $t$, for
sufficiently many distinct values of $k$ modulo $t$.
We apply this result to studying an analogue of the power generator
over an elliptic curve. These results are elliptic curve analogues
of those obtained for multiplicative groups of finite fields and
residue rings.
Categories:11L07, 11T23, 11T71, 14H52, 94A60 |
5. CJM 2001 (vol 53 pp. 414)
Nombres premiers de la forme $\floor{n^c}$ For $c>1$ we denote by $\pi_c(x)$ the number of integers $n \leq x$
such that $\floor{n^c}$ is prime. In 1953, Piatetski-Shapiro has
proved that $\pi_c(x) \sim \frac{x}{c\log x}$, $x \rightarrow +\infty$
holds for $c<12/11$. Many authors have extended this range, which
measures our progress in exponential sums techniques.
In this article we obtain $c < 1.16117\dots\;$.
Categories:11L07, 11L20, 11N05 |