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        riddles >> medium >> Difference of powers
        
(Message started by: NickH on May 1^st, 2003, 3:18pm)

Title: Difference of powers
Post by NickH on May 1^st, 2003, 3:18pm

Find all ordered pairs (m,n) of positive integers such that |3^m - 2ⁿ| < 10.

Title: Re: Difference of powers
Post by Leonid Broukhis on Jul 8^th, 2003, 9:14am

(1,2)
(2,1)
(2,2)
(1,3)
(2,3)
(2,4)
(3,5)

What did I miss?

Title: Re: Difference of powers
Post by towr on Jul 8^th, 2003, 9:55am

on 07/08/03 at 09:14:23, Leonid Broukhis wrote:

What did I miss?

(1,1)

Title: Re: Difference of powers
Post by Leonid Broukhis on Jul 8^th, 2003, 10:30am

That was a stupid mistake: I started listing pairs from (0,0),
then realised that m and n must be positive, and cut a group of lines that contained 0. (1,1) must have been somewhere among them. But really, why is it medium? Because of (3,5)? Or because you need to prove that there are no more? Well, continuous fractions of log(3)/log(2) help to do that.

Title: Re: Difference of powers
Post by NickH on Jul 9^th, 2003, 12:45pm

I placed it in medium because of the need to prove there are no more.

Please could you expand on the continued (?) fractions technique?

Title: Re: Difference of powers
Post by Leonid Broukhis on Jul 9^th, 2003, 5:21pm

After some thought, maybe it doesn't help to prove that there are no more, only to make me convinced about it. :)