Designing an Algorithmic Proof of the Two-Squares Theorem

Abstract

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 communitywe use Euclid's algorithm as an interface to investigate which numbers can be written as sums of two positive squares. The precise formulation of the problem as an algorithmic problem is the key, since it allows us to use algorithmic techniques and to avoid guessing. The notion of invariance, in particular, plays a central role in our development: it is used initially to observe that Euclid's algorithm can actually be used to represent a given number as a sum of two positive squares, and then it is used throughout the argument to prove other relevant properties. We also show how the use of program inversion techniques can make mathematical arguments more precise.

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

“…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.…”

“…We use the results presented in [14] as a starting point. In that paper, an extended version of Euclid's algorithm is inverted to investigate when a number can be written as the sum of two squares.…”

“…Euclid's algorithm is expressed in matrix terms 3 and computes a matrix D that is a product of Ls and Rs. The main theorem of [14] states that a number m at least 2 can be written as the sum of two squares if there is a number n such that 0<n<m and n 2 ∼ = −1 (mod m). Moreover, when the inputs to the algorithm satisfy these conditions, the final value of D is of the form M × L with M = M T .…”

“…Moreover, when the inputs to the algorithm satisfy these conditions, the final value of D is of the form M × L with M = M T . Although the matrices are key to establish the result in [14], no connection was established with the Stern-Brocot and Eisenstein-Stern trees. We investigate the connection in this section.…”

Αλγοριθμικές Διαδικασίες Ως Κύρια Μαθηματική Γνώση

“…Είναι γνωστό στη Θεωρία Αριθμών ότι κάθε πρώτος αριθμός p, με p  1 (mod 4), δηλαδή της μορφής p = 4k+1, είναι άθροισμα δυο τετραγώνων. Το θεώρημα αυτό είναι συνδεδεμένο με μεγάλους μαθηματικούς, όπως ο Διόφαντος, ο Fermat, αλλά και ο Euler που το απέδειξε (Ferreira, 2010a). Για την εύρεση των α και β τέτοιων, ώστε α 2 +β 2 = p, υλοποιείται ένας αλγόριθμος δύο βημάτων (Wagon, 1990).…”