Thanks, David -- those are my first steps into graphs and indeed, since I've posted my mail, I've discovered that those questions are quite difficult to answer. Thanks also for providing me the correct words « tree », « child », etc. Thanks to Gottfried Helms, too. Best, É. -----Message d'origine----- De : math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com [mailto:math-fun-bounces+eric.angelini=kntv.be@mailman.xmission.com] De la part de David Wilson Envoyé : lundi 9 octobre 2006 16:31 À : math-fun Objet : Re: [math-fun] Prime path Empirical evidence suggests that there is a prime on [(n^2-3n+8)/2, (n^2-n+4)/2] for every n >= 2, that is to say, every node of your tree has a prime child, so that there is an infinite path of increasing primes starting not only at 1 (counting 1 as prime), but at any node of your tree. On the other hand, Opperman's similar 1882 conjecture that there is a prime on [n^2, (n+1)^2] for every n >= 2 is also supported by empirical evidence, but resists proof to this day. This attests to the difficulty of such questions. Admittedly, your initial conjecture is weaker than the conjecture that every node of your tree has a prime child. We might be able to prove that most of the nodes of your tree have a prime child, and that we are able to choose a path that avoids those few nodes that do not have prime children. Still, such a proof would be very involved, well beyond my capabilities. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun