"Nice" Rational Functions

We consider simple rational functions $R_{mn}(x)=P_m(x)/Q_n(x)$, with $P_m$ and $Q_n$ polynomials of degree $m$ and $n$ respectively. We look for "nice" functions, which we define to be ones where as many as possible of the roots, poles, critical points and (possibly) points of inflexion are integer or, at worst, rational

On a problem of John Leech

AbstractWe consider the problem of finding two integer right-angled triangles, having a common base, where the respective heights are in the integer ratio n:1. By considering an equivalent elliptic curve problem, we find parametric solutions for certain values of n. Extensive computational resources are then employed to find those integers which do appear with 2⩽n⩽999

Measurements of FEL dynamics

SIGLEAvailable from British Library Document Supply Centre-DSC:DXN024390 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Small-degree parametric solutions for degree 6 and 7 ideal multigrades

We derive parametric solutions for 6 and 7 term ideal multigrades. These are of significantly smaller degree than previous solutions, such as those of Chernick

Two extreme Diophantine problems concerning the perimeter of Pythagorean triangles

We consider two Diophantine problems involving the perimeter of Pythagorean triangles. One has an enormous solution with the sides of the triangle > 101^15, whilst the other has no solution which answers a speculation of Frenicle de Bress