Re: exponential diophantine equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg62969] Re: exponential diophantine equations*From*: mike_in_england2000 at yahoo.co.uk*Date*: Fri, 9 Dec 2005 06:19:39 -0500 (EST)*References*: <dn8qs9$hj7$1@smc.vnet.net><dn95f6$kna$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi Andrea No problem - mistakes happen. Now I don't know much about diophantine equations (but would like to know more) so I had a look at your problem. The main thing that I know about them is that they are very hard to solve and that there is no algorithm that can solve general diophantine equations. With this in mind it is not suprising that Mathematicas standard commands cannot find a solution. The approach I took is one of brute force and ignorance . I define a function - dioph[x,k] that corresponds to your expression. I then loop over a large number of values for x and k and test the result. Every pair of x,k values that satisfies your equation gets added to the list: dioph[x_, k_] := (5 x)^2 - 2^k*3*(5 + k)^2 - 131*k - 7 Reap[ Do[ Do[ If[dioph[x, k] == 0, Sow[{x, k}];] , {x, 0, 1000} ] , {k, 0, 1000} ]] gave the output {Null, {{{31, 6}}}} so this looked at 1002001 possible x,k pairs and found the solution x=31, k=6. This took about 16 seconds on my machine. This is crude and could probably be vastly improved but at least it finds solutions. Recently I have been curious about diophantine equations. Where did this particular one come from? Why do you want to solve it? Thanks, Mike