Abstract

We show a new and constructive proof of the two-squares theorem, based on a somewhat unusual but very effective, way of rewriting the so-called extended Euclid's algorithm. Rather than simply verifying the result---as it is usually done in the mathematical community---we use Euclid's algorithm as an interface to investigate which numbers can be written as sums of two squares. The precise formulation of the problem as an algorithmic problem is the key, since it allows us to use algorithmic techniques and ...

Abstract

MathIS is a new project that aims to reinvigorate secondary-school mathematics by exploiting insights of the dynamics of algorithmic problem solving. This paper describes the main ideas that underpin the project. In summary, we propose a central role for formal logic, the development of a calculational style of reasoning, the emphasis on the algorithmic nature of mathematics, and the promotion of self-discovery by the students. These ideas are discussed and the case is made, through a number of examples that show ...

Abstract

Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid's algorithm. We illustrate how to use the algorithm as a verification interface (i.e., how to verify theorems) and as a construction interface (i.e., how to investigate and derive new theorems).The theorems that we verify are well-known and most of them are included in standard number-theory books. ...

Note (first note only)

Submitted for publication. Available at http://joaoff.com/publications/2009/euclid-alg/

Abstract

We derive an algorithm that enables the rationals to be efficiently enumerated in two different ways. One way is known and is credited to Moshe Newman; it corresponds to a deforestation of the so-called Calkin-Wilf tree of rationals. The second is new and corresponds to a deforestation of the Stern-Brocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In this way, we construct a Stern-Brocot enumeration algorithm with the same time and space complexity as ...