On Euclid’s algorithm and elementary number theory

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. 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.

A calculational approach to path-based properties of the Eisenstein–Stern and Stern–Brocot trees via matrix algebra

“…For reasons of historical accuracy, we deviate from common practice and use a different name for what is commonly know as Calkin-Wilf tree. As pointed out in [6], Stern [7] had already documented essentially the same structural characterisation of the rationals almost 150 years earlier than Calkin and Wilf. Stern attributes the structure to Eisenstein, so henceforth we refer to the "Eisenstein-Stern" tree of rationals where recent publications would refer to the "Calkin-Wilf" tree of rationals.…”

“…criteria based on parity as in the example given for Pascal's triangle). This paper is part of an endeavour which aims at reinvigorating mathematical content by adopting a calculational style of reasoning [6,19,14,20,21]. As suggested by the results shown in [22], the calculational method can indeed have a positive impact on mathematics education.…”

“…In fact, as described in [6], this property is known since at least 1858, since it is obviously present in Stern's paper [7]. An inductive proof can be found in [12] and an alternative proof based in branching can be found in [13]; both proofs are decomposed into three cases.…”

“…Stern attributes the structure to Eisenstein, so henceforth we refer to the "Eisenstein-Stern" tree of rationals where recent publications would refer to the "Calkin-Wilf" tree of rationals. For more details on Stern's characterisation, see the appendix in [6]. For a comprehensive account of properties of the SternBrocot tree, including further relationships with Euclid's algorithm, see [8, pp.…”

“…For the full details, we refer the reader to [5,6]. As discussed in the introduction, the matrix formulation has several advantages.…”