Re: how to solve the integer equation Abs[3^x-2^y]=1
- To: mathgroup at smc.vnet.net
- Subject: [mg103022] 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 03:15:32 -0400 (EDT)
- References: <200909031110.HAA24198@smc.vnet.net> <h7pl2g$gfb$1@smc.vnet.net>
As your prove is so superb and clear , now, I can easly prove that
the only solutions of 3^x-2^y==1 are (1,1) and (2,3).
for 3^x==(4-1)^x==1+2^y,
(-1)^x==1+2^y ( mod 4),
when x=2a+1, it must be y<2; when x=2a, y>1 and (3^a+1)(3^a-1)=2^y,so
only x=2.
Proved!
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?
- 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