Decimal periods and their tables: A German research topic (1765–1801)

Bullynck, Maarten

doi:10.1016/j.hm.2008.09.004

Cited by 20 publications

(9 citation statements)

References 6 publications

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“…. , 9}, was investigated by Gauss and earlier authors centuries ago, see [5] for a historical account, and [33] for recent developments. As discussed in Articles 14-18 in [17], the period, denoted by ord p (10) = d ≥ 1, is a divisor of p − 1.…”

Section: Application To Repeated Decimalsmentioning

confidence: 99%

Densities Of Primes And Primitive Roots

Carella

2017

Preprint

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Let u = ±1, v 2 be a fixed integer, let p ≥ 2 be a prime, and let ord p (u) = d|p−1 be the order of u mod p. This note provides an effective lower bound #{p ≤ x : ord p (u) = p − 1} ≫ x(log x) −1 for the number of primes p ≤ x with a fixed primitive root u mod p for all large numbers x ≥ 1. The current results in the literature have the lower bound #{p ≤ x : ord p (u) = p − 1} ≫ x(log x) −2 , and restrictions on the fixed primitive root to a subset of at least three or more integers.

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Section: Application To Repeated Decimalsmentioning

confidence: 99%

Densities Of Primes And Primitive Roots

Carella

2017

Preprint

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“…However, the generalization to complex variables is straightforward and doesn't change our conclusions. For that purpose we use the Bézout's identity [29] stating:…”

Section: Non-harmonic Revival Of Expectation Valuesmentioning

confidence: 99%

Quantum revivals and magnetization tunneling in effective spin systems

et al. 2016

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Quantum mechanical objects or nano-objects have been proposed as bits for information storage. While time-averaged properties of magnetic, quantum-mechanical particles have been extensively studied experimentally and theoretically, experimental investigations of the real time evolution of magnetization in the quantum regime were not possible until recent developments in pump-probe techniques. Here we investigate the quantum dynamics of effective spin systems by means of analytical and numerical treatments. Particular attention is paid to the quantum revival time and its relation to the magnetization tunneling. The quantum revival time has been initially defined as the recurrence time of a total wave-function. Here we show that the quantum revivals of wave-functions and expectation values in spin systems may be quite different which gives rise to a more sophisticated definition of the quantum revival within the realm of experimental research. Particularly, the revival times for integer spins coincide which is not the case for half-integer spins. Furthermore, the quantum revival is found to be shortest for integer ratios between the on-site anisotropy and an external magnetic field paving the way to novel methods of anisotropy measurements. We show that the quantum tunneling of magnetization at avoided level crossing is coherent to the quantum revival time of expectation values, leading to a connection between these two fundamental properties of quantum mechanical spins.

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“…. , 9}, was investigated by Wallis, Lambert, Gauss and earlier authors centuries ago, see [10] for a historical account. As discussed in Articles 14-to-18 in [43], the period, denoted by ord p (10…”

Section: Densities For the Repeating Decimalsmentioning

confidence: 99%

Asymptotic For Primitive Roots Producing Polynomials

Carella¹

2016

Preprint

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Let x ≥ 1 be a large number, let f (n) ∈ Z[x] be a prime producing polynomial of degree deg(f ) = m, and let u = ±1, v 2 be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes p = f (n) ≤ x with a fixed primitive root u is derived in this note. This asymptotic result has the form, where c(u, f ) is a constant depending on the polynomial and the fixed integer. Furthermore, new results for the asymptotic order of elliptic primes with respect to fixed elliptic curves E : f (X, Y ) = 0 and its groups of F p -rational points E(F p ), and primitive points are proved in the last chapters.

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Decimal periods and their tables: A German research topic (1765–1801)

Cited by 20 publications

References 6 publications

Densities Of Primes And Primitive Roots

Densities Of Primes And Primitive Roots

Quantum revivals and magnetization tunneling in effective spin systems

Asymptotic For Primitive Roots Producing Polynomials

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