A diophantine equation for sums of consecutive like powers

We show that the diophantine equation $n^\ell+(n+1)^\ell + ...+ (n+k)^\ell=(n+k+1)^\ell+ ...+ (n+2k)^\ell$ has no solutions in positive integers $k,n \ge 1$ for all $\ell \ge 3$.Comment: History of the problem adde

Effect of Hydrostatic Guideway on Reduction of Machine Tool Ram Vibration

The guideways provide translational movement of machine parts and have a major impact on the resulting utility properties of the ma-chine tools, such as machining accuracy, surface quality and productivity. It is generally understood that hydrostatic guideways have better damping properties than linear guideways with rolling elements. However, quantitative expressions of better damping appear in the literature very sporadically. Therefore, this paper aims to compare hydrostatic and linear guideways and to assess the impact of higher damping on a ram vibrations of a large machine tool. The forced oscillations amplitude of the ram tool center point was calculat-ed by the FEM model of the deformable ram and stiffness and damping model of guideways. Results indicate that hydrostatic guide-way reduce the forced oscillation amplitude of the first eigenfrequency 15 times in case of the modeled machine tool

Max Dehn, Axel Thue, and the Undecidable

This is a short essay on the roles of Max Dehn and Axel Thue in the formulation of the word problem for (semi)groups, and the story of the proofs showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve