Abstract We provide elementary explanations in terms of palindromic continued fraction expansions for the factorization of integers of the form a2 + 1, including Fermat numbers and Cunningham project numbers. This provides a generalization and more complete explanation of the factorization of the sixth Fermat number given by Freeman Dyson at the turn of the century. This explanation may provide...

Let θ = [0; a1, a2, . . .] be an algebraic number of degree at least three. Recently, we have established that the sequence of partial quotients (a`)`≥1 of θ is not too simple and cannot be generated by a finite automaton. In this expository paper, we point out the main ingredients of the proof and we briefly survey earlier works.

This paper introduces a new algorithm for the evaluation of monomials in two variables X " JJ~ based upon the continued fraction expansion of a/b. A method for fast explicit generation of addition chains of small length for a positive integer n is deduced from this Algorithm. As an illustration of the properties of the method, a Schoh-Brauer-like inequality p(N) s nb + k + p(n + l), is shown to...