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        riddles >> medium >> Subtract The Expressions, Get 1
        
(Message started by: K Sengupta on Jun 25^th, 2008, 8:45am)

Title: Subtract The Expressions, Get 1
Post by K Sengupta on Jun 25^th, 2008, 8:45am

Determine all possible triplet(s) (P, Q, R) of positive integers, that satisfy this equation:

(P+2)^R+1 - (P+1)^Q+1 = 1

Title: Re: Subtract The Expressions, Get 1
Post by TenaliRaman on Jun 30^th, 2008, 1:11pm

Well, it may be futile, but my working so far has given me,
Q > R >= (2Q - 1) / 3
P = ((R+1) / (Q - R)) - 1

No, it doesn't mean that if P,Q,R satisfy these requirements, they are the solution. On the contrary, if they are the solution to the above system, they would satisfy the above equations :( (Sad, I know).

I am trying to get some headway with this, but so far all efforts were in vain. Just posted it to see if anyone else can make any sense of it.

-- AI

Title: Re: Subtract The Expressions, Get 1
Post by towr on Jun 30^th, 2008, 1:37pm

[hide]"Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference [between powers>2 of integers m], between 8 and 9".

So (1,2,1)[/hide]
[hide]linky to sequence A001597[/hide] (http://www.research.att.com/~njas/sequences/A001597)

Actually, [hide]Should this even be in medium? Cause it's little more than "guess the theorem" I doubt anyone here can be expected to prove it.[/hide]

Title: Re: Subtract The Expressions, Get 1
Post by Eigenray on Jun 30^th, 2008, 7:25pm

Note that we can factor this in more ways if [hide](Q+1) is odd or (R+1) is even[/hide]. Actually, these both must be the case.

on 06/30/08 at 13:37:14, towr wrote:

Actually, [hide]Should this even be in medium? Cause it's little more than "guess the theorem" I doubt anyone here can be expected to prove it.[/hide]

But this is only a special case of Catalan's conjecture (see [link=http://www.ocf.berkeley.edu/%7Ewwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1184647059]here[/link] for another one). There is an elementary proof.