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For any finite Galois extension $K$ of $\mathbb{Q}$ and any conjugacy class $C$ in $\text{Gal}\left( {K}/{\mathbb{Q}}\; \right)$, we show that there exist infinitely many Carmichael numbers composed solely of primes for which the associated class of Frobenius automorphisms is $C$. This result implies that for every natural number $n$ there are infinitely many Carmichael numbers of the form ${{a}^{2}}\,+\,n{{b}^{2}}$ with $a,\,b\,\in \,\mathbb{Z}$.
We give new bounds on sums of the form ∑ n≤NΛ(n)exp (2πiagn/m) and ∑ n≤NΛ(n)χ(gn+a), where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums ∑ p≤Nexp (2πiaMp/m) and ∑ p≤Nχ(Mp) with Mersenne numbers Mp=2p−1, where p is prime.
Assuming a conjecture intermediate in strength between one of Chowla and one of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m≥1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.
where $g$ and $n$ are positive integers, $n$ is composite, and $P\left( n \right)$ 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 $\left\{ {{f}_{g}}(n) \right\}$ and $\left\{ {{h}_{g}}(n) \right\}$ 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 ${{\tilde{f}}_{g}}(n)=g\,{{f}_{g}}(n)$ and ${{\tilde{h}}_{g}}(n)=g{{h}_{g}}(n)$.
Let $\varphi(\cdot)$ be the Euler function and let $\sigma(\cdot)$ be the sum-of-divisors function. In this note, we bound the number of positive integers $n\le x$ with the property that $s(n)=\sigma(n)-n$ divides $\varphi(n)$.
Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n > 1. We show that, for every real number , there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c(η) = 1+ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P(q − a) ≍ qθ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisor of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ P(q - a) for a > 0, and show that for infinitely many primes q, this map can be iterated at least (log logq)1+o(1) times before it terminates.
We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p½+ε ≤ N ≤ p, then among the first N elements of ℬα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.
Let $p$ be a prime and $\vartheta$ an integer of order $t$ in the multiplicative group modulo $p$. In this paper, we continue the study of the distribution of Diffie–Hellman triples$(\vartheta^x, \vartheta^y, \vartheta^{xy})$ by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie–Hellman triples as the pair $(x,y)$ varies over small boxes.
By
William C. Banks, Professor Maxwell School of Citizenship and Public Affairs Syracuse University,
Alejandro D. Carrió, Visiting Professor of Law Louisiana State University; Professor of Law Palermo University
Argentine society has experienced terrorism from domestic and external sources since the late 1960s. Domestic terrorism appeared mainly in the 1970s, first as leftist guerrilla movements that sought to attract attention when they failed to field an organized political movement. This small band of terrorists effectively paved the way for the appearance of brutal forms of state terrorism. The repressive state practices were first justified as a necessary antidote to the guerrilla activities, but quickly expanded into a blunt and massive campaign to eliminate any dissent to military rule. Unfortunately, the extra-legal methods employed by the Argentine government for fighting terrorist violence were mostly condoned or acquiesced in by the judiciary, and the legacy of the ‘Dirty War’ waged by the military junta continues to haunt the Argentine courts.
In more recent times international terrorism appeared. A massive attack destroyed the Embassy of Israel in Buenos Aires in 1992. In 1994, the building that harboured the two most prominent Jewish organizations in Argentina was targeted with another deadly bomb. In the combined attacks, conventional explosives caused in the aggregate about one hundred deaths and hundreds of injuries. In both cases criminal investigations have been ongoing for more than ten years, although there has been little success in bringing those responsible to justice. After one president was accused of interfering with the investigations of the bombings, a new president has promised to invigorate the investigation and to make its findings public.
Let $\varphi(\cdot)$ denote the Euler function, and let $a>1$ be a fixed integer. We study several divisibility conditions which exhibit typographical similarity with the standard formulation of the Euler theorem, such as $a^n \equiv 1\!\!\!\!\pmod{\varphi(n)}$, and we estimate the number of positive integers $n\le x$ satisfying these conditions.
In late 2001, investigators excavated a solitary Middle Archaic burial from the Plains-Prairie border in east-central Kansas. The burial was contained in a dissected colluvial apron at the foot of the valley wall, in a soil horizon that began accumulating around 9000 B.P. Burial goods include deer bone, a drill, and a side-notched projectile point/knife, the morphology of which is consistent with side-notched Middle Archaic points of the North American Central Plains and Midwest. Use-wear analysis shows that the stone tools were used before being placed with the burial and were not manufactured specifically as burial goods. A radiocarbon assay of the deer bone in direct association with the burial yielded a radiocarbon age of 6160 ± 35 B.P. This is one of only a few burials older than 5,000 years in the region. Comparison of this burial to other coeval regional burials shows similarities in burial practices.
We give estimates for exponential sums of the form $\sum_{n \leq N}\Lambda(n)\exp(2 \pi i a g^n/m)$, where m is a positive integer, a and g are integers relatively prime to m, and $\Lambda$ is the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form $\sum_{p \leq N}\exp(2 \pi i a M_p/m)$, where Mp is the Mersenne number; $M_p=2^p-1$ for any prime p. We also estimate some closely related sums, including $\sum_{n \leq N}\mu(n)\exp(2 \pi i a g^n/m)$ and $\sum_{n \leq N}\mu^2(n)\exp(2 \pi i a g^n/m)$, where $\mu$ is the Möbius function.
We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring ${{\mathbb{F}}_{q}}\left[ x \right]\,/\,M\left( x \right)$ for collections of polynomials either of the form ${{f}^{-1}}{{g}^{-1}}$ or of the form ${{f}^{-1}}{{g}^{-1}}\,+\,afg$, where $f$ and $g$ are polynomials coprime to $M$ and of very small degree relative to $M$, and $a$ is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.
Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field 𝔽p is isomorphic to a curve y2 = x3 + ax + b with a and b of size O (p¾). In this paper, we show that almost all elliptic curves satisfy the stronger bound O (p⅔). The problem is motivated by cryptographic considerations.
Individuals infected with Human Immunodeficiency Virus (HIV) and having cognitive impairment have been described as having slow mentation. Data supporting this proposition come from a variety of sources, including Sternberg's (1966) item recognition memory task. The procedure nominally provides an index of speed of mental operations, independent from input/output demands. However, since the original use of this procedure in the 1960s, advances in cognitive psychology have revealed many of its limitations. The purpose of the present study was to examine the psychometric characteristics of this task. Each participant performed the Sternberg item recognition task twice, 6 mo apart. The stability of the estimate of the slope of regression equations and for zero intercept ranged from excellent (r = .87) to poor (r = .30), and the data from many individual subjects could not be reliably modelled using multiple linear regression techniques. These data, as well as those from previous research, demonstrate the limited practical use of this task in clinical samples. Furthermore, as cognitive psychological theory has advanced in the past 30 yr, the conceptual underpinnings of the procedure have essentially evaporated. (JINS, 1995, 1, 3–9).
Dietary manipulation was used to produce a similar series of milks from both Friesian and Jersey cows. The gross compositions of the milks, the fatty acid (FA) composition of the milk fats, the distribution of molecular sizes in the triglycerides of the milk fat, the melting properties of the milk fats, and the whipping properties of creams containing 360 and 400 g fat/kg were measured. Changes in gross composition and FA composition were as expected from the use of dietary oil supplements, but it was established that the mathematical relation between 18:0 and 18:1 differed between breeds, the Jersey yielding a milk fat with a lower proportion of 18:1 for a given value of 18:0. Control diets free from added fat produced milk fats with essentially unimodal triglyceride distributions, whereas fatrich diets produced bimodal distributions. The slight differences in these distributions between breeds were merely a reflection of variations in FA composition rather than in synthetic procedures. Differences in the whipping properties of creams containing 360 and 400 g fat/kg were consistent with literature observations. Dietary manipulation had little effect on the whipping properties of creams derived from Friesian cows, but caused considerable changes in the corresponding properties of the creams from Jersey cows. The only property that behaved similarly in the creams from the two breeds was the butter time, i.e. the time taken for butter granules to form on prolonged whipping of the cream. A major determinant of the butter time appeared to be the proportion of the fat that was molten at the temperature at which the whipping experiments were carried out.
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