Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)

*To*: mathgroup at smc.vnet.net*Subject*: [mg57300] Re: [mg57278] Solve or Reduce on a monstrosity of an expresssion (and a prize!)*From*: Daniel Reeves <dreeves at umich.edu>*Date*: Tue, 24 May 2005 05:12:37 -0400 (EDT)*References*: <200505230620.CAA04045@smc.vnet.net> <EEA0FBBD-9C31-419A-8D0B-7C73AE4DA32E@akikoz.net> <D9EDDB86-15A8-4865-AF0E-9F89803AE6F2@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

Andrzej, thanks so much! I'm still stumped but I did just put together overwhelming empirical evidence that your conjecture is dead on. For all n there are 2 roots: a negative one, -(n+1)^2 / (n (n-1)) and the positive one, (n-1)/n. The prize is officially being augmented by an order of magnitude! > I still have not got a proof, but thinking about the above proof for > the case n=2 some conjectures come to my mind which might just > possible point tothe the way to the general proof. The main idea is > Daniel Reeves' one: to show that D[f[x,n],x] is not zero on (0,(n-1)/ > 2). If we look at the above argument for n=2 we see that it amounts > the fact that D[f[x,n],x]==0 has just two rots, one of which is at > (n-1)/n and the other is negative. One is tempted to conjecture that > the same is true for any n: the derivative has just two roots, one is > at (n-1)/n and the other is negative (I have no good guess at this > time as to where it might be, in the n=2 example it is at -9/2). > > At least for one part of the conjecture there is some confirmation: > > > D[f[x,3],x]/.x->2/3//FullSimplify > > 0 > > > FullSimplify[D[f[x, 4], x] /. x -> (4 - 1)/4] > > 0 > > although again the general n seems to take for ever. There may also > be a non-computational proof that the equation can only have two > roots and that one of them must be negative. I will think about it > more when I have a little more free time but at the moment i view > these only as long shot conjectures. > > Andrzej -- http://ai.eecs.umich.edu/people/dreeves - - google://"Daniel Reeves" "Math is like love: a simple idea but it can get complicated." --R.Drabek

**References**:**Solve or Reduce on a monstrosity of an expresssion (and a prize!)***From:*Daniel Reeves <dreeves@umich.edu>