Re: exponential diophantine equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg62934] Re: exponential diophantine equations*From*: dh <dh at metrohm.ch>*Date*: Thu, 8 Dec 2005 06:25:38 -0500 (EST)*References*: <dn8qs9$hj7$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi Andrea see below, Daniel Andrea wrote: > I'm trying to find out how to use Mathematica to find solutions to > exponential diophantine equations like the following: > > (5x)^2 - 2^k * 3 * (5+k)^2 - 131 * k + 7 = 0. I want to obtain > solutions for x and k. (One solution is x = 31, k = 6, but I didn't find > this using Mathematica!) You did not find it because it is not a solution: (5x)^2 - 2^k*3*(5 + k)^2 - 131*k + 7 /. {x -> 31, k -> 6} gives 14 and not zero. There is no solution as you may convince yourself: (5x)^2 == -(- 2^k*3*(5 + k)^2 - 131*k + 7) the left side is always positive, therefore, we only need to consider k for which the right side is positive, that is k>=0. In this region the left side is always smaller than the right side that grows exponentially. Daniel > > I'm not very skilled in Mathematica. I tried Solve and Reduce but I just > get either > > The equations appear to involve the variables to be solved for in an > essentially non-algebraic way. > or > This system cannot be solved with the methods available to Reduce. > > I'm trying to find a way to get at least one or two solutions. Is there a > way in Mathematica to do this? I know that exponential ones are not easy... > > Thanks for any help. > > Andrea >