On 16 Jul, 06:13, Gottfried Helms <he...@uni-kassel.de> wrote: > Am 15.07.2007 05:30 schrieb arithmeticae: > > > If you really like to analyze the most simple high-order root-solving > > algorithms then you should take a look at: > > >http://mipagina.cantv.net/arithmetic/rmdef.htm > > > It is striking to realize that these new extremely simple artihmetical > > algorithms do not appear in any text on numbers since Babylonian times > > up to now. > > Yes, I'd second that. It surprises me, that this method is > not more widely discussed. It is -at least- an amazing > approach in his simpliness and in its line of proceeding, > even if it should not be efficient. > > Hope, it will make its way into some books, at least as > an annotation, or in journals/books which are dedicated > to recreational and surprising mathematics.
I don't think the claim that these methods are in any way new stands up to scrutiny. The idea of Farey dissections is clearly not new. It is mentioned in Hardy and Wright for example. Hurwitz wrote a paper "Ueber die Irrationalzahlen" in the 1890s which describes a "mediant" method based on Farey fractions that produces best rational approximations. Monkmeyer and Mahler have examined generalizations of Farey fractions, essentially a higher order "mediant" method, intended to produce best rational simultaneous approximations to a set of irrationals. Can Morin find best rational approximations to cubrt(2), cubrt(4) with his methods ? He has not been able to do so in the past.