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• Equivalence and range of quadratic forms

  123-129

  Sándor Szabó

  Views:

  10

  If two quadratic forms are equivalent, that is, if there is a linear transformation with integer coefficients and determinant 1 or −1 which takes one form to the other, then their ranges are the same and also their determinants are the same. The result of the paper is that for positive definite binary quadratic forms the converse is also true. Namely, if two positive definite binary quadratic forms of the same determinant have the same range, then they are equivalent. The arguments are guided by geometric considerations.

• On four-dimensional crystallographic groups

  391-404

  Eszter Horváth

  Views:

  9

  In his paper [12] S. S. Ryshkov gave the group of integral automorphisms of some quadratic forms (according to Dade [6]). These groups can be considered as maximal point groups of some four-dimensional translation lattices in E^4. The maximal reflection group of each point group, its fundamental domain, then the reflection group in the whole symmetry group of the lattice and its fundamental domain will be discussed. This program will be carried out first on group T. G. Maxwell [9] raised the question whether group T was a reflection group. He conjectured that it was not. We proved that he had been right. We shall answer this question for other groups as well. Finally we shall give the location of the considered groups in the tables of monograph [4]. We hope that our elementary method will be useful in studying linear algebra and analytic geometry. Futhermore, 4-dimensional geometry with some visualisation helps in better understanding important concepts in higher-dimensional mathematics, in general.

• Solving Diophantine equations with binomial coefficients in study group sessions using both elementary and higher mathematical methods

  1-12

  Richárd Rakamazi

  Views:

  28

  The paper can be considered as the continuation of [4] in the sense that we are studying Diophantine equations containing binomial coefficients. It was an important aspect that one should be able to discuss these problems — even if not in complete depth — also in high school study group sessions with the most talented students. We present various methods through several examples, which help the successful handling of other questions too, including problems in math competitions. Our discussion starts with the elementary treatment of easier problems, and then proceed gradually to more difficult questions which require higher mathematical methods.