Re: how to solve the integer equation Abs[3^x-2^y]=1
- To: mathgroup at smc.vnet.net
- Subject: [mg103023] Re: how to solve the integer equation Abs[3^x-2^y]=1
- From: a boy <a.dozy.boy at gmail.com>
- Date: Fri, 4 Sep 2009 03:15:43 -0400 (EDT)
- References: <200909031110.HAA24198@smc.vnet.net> <h7pl2g$gfb$1@smc.vnet.net>
On Sep 4, 7:56 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> On 3 Sep 2009, at 13:10, a boy wrote:
>
> > Does the equation |3^x-2^y|=1 give only 4 groups of solution?
> > (x,y)= (0,1),
> > (1,1),
> > (1,2),
> > (2,3)
>
> > can anyone give any else solution?
> > when the two integers x and y become bigger and bigger, is there a
> > pair integer (x,y) to give a small value for |3^x-2^y|? Or else,how
> > to prove the equation |3^x-2^y|=1having only 4 groups of integer
> > solution?
>
> Here is the solution to one half of your problem, showing that the
> only integer solutions of the equation 2^y-3^x == 1 are (0,1) and
> (1,2). The other half of the problem is to show that the only
> solutions of 3^x-2^y==1 are (1,1) and (2,3). The proof should be
> similar "in spirit", but it seems harder so I will leave it to you.
>
> So, consider the equation 2^y-3^x == 1. For x==0, we must have y==1.
> Clearly, we can't have y==0. Suppose both x and y >= 1. Since 2^y ==
> (3-1)^y == (-1)^y mod 3 and 3^x + 1 == 1 mod 3, y must be even. Let y
> = 2a. Then 2^(2a)-1 == 3^x, hence (2^a-1)(2^a+1)==3^x. This is only
> possible if both factors are powers of 3, i.e. 2a-1==3^u and 2a+1==3^v
> (where u,v>=0). Hence 3^v-3^u == 2. If both u and v >=1 then the left
> hand side is divisible by 3, a contradiction. Therefore v==1 and u==0.
> Since u+v == x, x must be 1, a==1, so y =2. So the only solutions are
> (0,1) and (1,2).
>
> Andrzej Kozlowski
And more,I wonder that
does it exist two infinite and increasing integer sequence {Xi}
and {Yi} to
satisfy {|3^Xi-2^Yi|} progressively decreasing?
Could you give me Yes or NO? and why?
Oh, the answer to this is to be No, for 1 is the last element of {|
3^Xi-2^Yi|} .
Now my question is changed to :
To construct an increasing integer pair sequence {(Xi,Yi)} satisfy
that
1) {Xi} is progressively increasing
2) {|3^Xi-2^Yi|} progressively decreasing
What is L=the maximum length of constructed sequence?
I think it's hard to me, can you give me a good solution?
- References:
- how to solve the integer equation Abs[3^x-2^y]=1
- From: a boy <a.dozy.boy@gmail.com>
- how to solve the integer equation Abs[3^x-2^y]=1