163 research outputs found
"Nice" Rational Functions
We consider simple rational functions , with
and polynomials of degree and 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
Computation of perfect "almost-cuboids"
We discuss generating parallelepipeds, with 4 rectangular faces, which have rational lengths and all face and space diagonals also rational
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
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