Effective primality tests for integers of the formsN=k3 n +1 andN=k2 m 3 n +1

Using third roots of unity Proth's theorem for primality testing is generalized to integers of the formN=k3 ⁿ +1, avoiding the use of Lucas sequences which are more suitable ifN+1 is factored instead ofN–1. This approach has the advantage of being easily combined with Proth's test and gives polynomial time algorithms for testing integers of the formN=k2 ^m 3 ⁿ +1.     

Wievielek-stellige Fibonaccizahlen gibt es?

Ohne Zusammenfassung     

Cavitation in superfluid helium-4 at low temperature

We have studied the nucleation of bubbles in pure superfluid helium-4 at temperatures down to 65 mK. We have found that the nucleation is a stochastic process, and that at temperatures below 600 mK the nucleation rate is independent of temperature. These results are consistent with the assumption that the nucleation takes place via quantum tunneling. Received: 15 November 1997 / Received in final form: 19 December 1997 / Accepted: 22 January 1998     

Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface

We propose a very simple method to determine the electrical tip-surface force in Atomic Force Microscopes used to study the electrical properties of metallic or insulating materials; the analysis of the measurements as well as determination of the appropriate experimental procedures requiring an analytical model of the tip-surface capacitance. The comparison of force expressions obtained by this method with those obtained by exact derivation in the case of the sphere-infinite plane system shows very good agreement. This method is then applied to determine the tip-surface force, the real shape of the tip being introduced in the derivation. The obtained expression is compared to experimental and numerical data. We emphasize that this method is very general and can be applied to any axially symmetric capacitor. Received: 17 June 1997 / Received: 16 September 1997 / Accepted: 7 November 1997     

Adsorption of He on He crystal surfaces

We have compared the surface properties of crystals respectively grown from normal ⁴He containing 130 ppb of ³He and from ultrapure ⁴He (0.4 ppb). Below 0.4 K, ³He impurities are found to decrease both the surface tension and the step energy. Our results are consistent with the existence of two-dimensional bound states for ³He atoms at the solid-liquid interface. Quantitative agreement with the data is found with a binding energy _s ≈ 4.3 K and a ³He density saturating around 0.4 monolayer. The presence of steps is found to increase the binding energy by about 10 mK.     

Intrachain and interchain triplet-triplet exciton annihilation in a quasi-one-dimensional crystal: 1,4-dibromonaphthalene (DBN)

The modulation of delayed fluorescence in 1,4-dibromonaphthalene by magnetic fields has been studied at room temperature. The analysis of high field resonance positions shows that, besides annihilation of triplet excitons in the same one-dimensional stack, annihiliation also occurs between triplets localized on different, neighbouring stacks, in particular those corresponding to the two molecules of the asymmetric pair.