Re: Solving Diophantine Equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg61241] Re: [mg61185] Solving Diophantine Equations*From*: Andrzej Kozlowski <andrzej at yhc.att.ne.jp>*Date*: Fri, 14 Oct 2005 05:53:26 -0400 (EDT)*References*: <200510120542.BAA09212@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On 12 Oct 2005, at 14:42, Diana wrote: > Math group, > > I am trying to start doing numerical solving of Diophantine > Equations. I > want to do extensive calculations, but the kernel aborts for lack > of memory. > > For example, the following command aborts midstream: > > Table[If[(x^3-1)/(x-1)==(y^n-1)/(y-1) && x!=y, > {{x,y,n},}],{x,2,1000},{y,2,1000},{n,1,1001,2}] > > I have tried doing many smaller steps, but then I have a very long > file of > multiple commands. > > Would you make a suggestion as to how to process these solving > commands with > large values of the variables? > > Thanks, > > Diana > -- > ===================================================== > "God made the integers, all else is the work of man." > L. Kronecker, Jahresber. DMV 2, S. 19. > > I doubt this will be of much help, but there are some obvious inequalities that must hold between the possible values of x and y, which cann reduce the number of possible cases that need to be looked at. For example, you must have x > y^((n - 1)/2) Since ifn this were not true the LHS of (x^3-1)/(x-1)==(y^n-1)/(y-1) would be too large. Similarly it is also obvious that, x < (n - 1)*(y^(n - 1)/2) These are very rough bounds and as I spent only about 1 minute looking for them you should be able to work out better ones. In this way you can at least reduce the number of x's that need to be looked at for a given y and n. Even so this looks like a pretty tough task to attampt on a present day computer. Andrzej Kozlowski Andrzej Kozlowski Tokyo, Japan

**References**:**Solving Diophantine Equations***From:*"Diana" <diana53xiii@earthlink.remove13.net>