Hesse Pencils and 3-Torsion Structures

This paper intends to focus on the universal property of this Hesse pencil and of its twists. The main goal is to do this as explicit and elementary as possible, and moreover to do it in such a way that it works in every characteristic different from three

Triangular Numbers and Elliptic Curves

Some arithmetic of elliptic curves and theory of elliptic surfaces is used to find all rational solutions (r, s, t) in the function field Q(m, n) of the pair of equations r(r + 1)/2 = ms(s + 1)/2 r(r + 1)/2 = nt(t + 1)/2. It turns out that infinitely many solutions exist. Several examples will be given

Further remarks on rational albime triangles

In this note we present further number theoretic properties of the rational albime triangles, in particular, the distribution of acute vs. obtuse rational albime triangles. The notion of albime triangle is extended to include the case of external angle bisector. The proportion of internal vs. external rational albime triangles is also computed<br/

The last chapter of the Disquisitiones of Gauss

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.Comment: 13 page

Albime triangles and Guy’s favourite elliptic curve

This text discusses triangles with the property that a bisector at one vertex, the median at another, and the altitude at the third vertex are concurrent. It turns out that since the 1930s, such triangles appeared in the problem sections of various journals. We recall their well known relation with points on a certain elliptic curve, and we present an elementary proof of the classical result providing the group structure on the real points of such an elliptic curve. A consequence of this answers a question posed by John P. Hoyt in 1991

Effects of the galactic magnetic field upon large scale anisotropies of extragalactic Cosmic Rays

The large scale pattern in the arrival directions of extragalactic cosmic rays that reach the Earth is different from that of the flux arriving to the halo of the Galaxy as a result of the propagation through the galactic magnetic field. Two different effects are relevant in this process: deflections of trajectories and (de)acceleration by the electric field component due to the galactic rotation. The deflection of the cosmic ray trajectories makes the flux intensity arriving to the halo from some direction to appear reaching the Earth from another direction. This applies to any intrinsic anisotropy in the extragalactic distribution or, even in the absence of intrinsic anisotropies, to the dipolar Compton-Getting anisotropy induced when the observer is moving with respect to the cosmic rays rest frame. For an observer moving with the solar system, cosmic rays traveling through far away regions of the Galaxy also experience an electric force coming from the relative motion (due to the rotation of the Galaxy) of the local system in which the field can be considered as being purely magnetic. This produces small changes in the particles momentum that can originate large scale anisotropies even for an isotropic extragalactic flux.Comment: 11 pages, 4 figure