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. The new results concern distributivity properties of the greatest common divisor and a new algorithm for efficiently enumerating the positive rationals in two different ways. One way is known and is due to Moshe Newman. 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 Newman's algorithm. A short review of the original papers by Stern and Brocot is also included.

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 Newman's algorithm.Keywords: Calkin-Wilf tree, Stern-Brocot tree, algorithm derivation, enumeration algorithm, rational numbers Recently, there has been a spate of interest in the construction of bijections between the natural numbers and the (positive) rationals (see [GLB06, KRSS03, CW00] and [AZ04, pages 94-97]). Gibbons et al [GLB06] describe as "startling" the observation that the rationals can be efficiently enumerated ½ by "deforesting" the Calkin-Wilf [CW00] tree of rationals. However, they claim that it is "not at all obvious" how to "deforest" the Stern-Brocot tree of rationals. (For information on the Stern-Brocot tree, see [GKP94,.)In this paper, we derive an efficient algorithm for enumerating the rationals both in Calkin-Wilf and Stern-Brocot order. The algorithm is based on a bijection between the rationals and invertible 2×2 matrices. The key to the algorithm's derivation is the * Funded by Fundação para a Ciência e a Tecnologia (Portugal) under grant SFRH/BD/24269/2005½ By an efficient enumeration we mean a method of generating each rational without duplication with constant cost per rational in terms of arbitrary-precision simple arithmetic operations.

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This paper presents JaSkel, a skeleton-based framework to develop parallel and grid applications. The framework provides a set of Java abstract classes as a skeleton catalogue, which implements recurring parallel interaction paradigms. This approach aims to improve code efficiency and portability. It also helps to structure scalable applications through the refinement and composition of skeletons. Evaluation results show that using the provided skeletons do contribute to improve both application development time and execution performance.