A086446 extension?

[James Buddenhagen]

----- Original Message ----- 
From: "David Wilson" <davidwwilson at comcast.net>
To: "Sequence Fans" <seqfan at ext.jussieu.fr>
Sent: Saturday, April 02, 2005 3:19 PM
Subject: A086446 extension?


> Can
> 
> http://maths.paisley.ac.uk/allanm/ecrnt/knight/knposres
> 
> be used to extend A086446?
> 
> - David W. Wilson

I would be a bit cautious.  Given positive integer N, the Diophantine 
equation N=(x+y+z)*(1/x + 1/y + 1/z) either has no integer solutions (x,y,z) 
or has infinitely many. (Perhaps there are some trivial exceptions).  

Often, e.g. N=14 some solutions have all of x,y,z positive and some 
solutions have some of x,y,z negative.  So one cannot immediately 
conclude from the presence of negative values in the table at your link 
that there will not also be solutions with x,y,z all positive.  

When N=14 some solutions are:  [x,y,z]=[3, 2, 10], [10, 3, 15], 
[715, -3619, 14664], [-7896, -385, 1560], [129609938, 665659395, 398619322], 
[1563392295, 508332555, 304406698], 
[3026000198480402499943863, 10873046781459417805417450, 2249722029555895456095675],
 [3053593877043831122496050, 4107252165654093769719898, 849823960453237978650627],
[-15624758450605286212397382416269320, -330144846448075478619791396166893880, 
100497528394365361363242440213278081], 
[721684452988315695119447152936023749, -112203209727214182895046727899978280, 
34155145466652403549506588347234611], ...

 From this I also conclude that sequences A102774, A102775, A102777 for values 
of x, y, z for N>=11 are not well-defined, at least as they are presently 
described.

Jim Buddenhagen