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Topic: Subtract The Expressions, Get 1 (Read 698 times) |
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K Sengupta
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Subtract The Expressions, Get 1
« on: Jun 25th, 2008, 8:45am » |
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Determine all possible triplet(s) (P, Q, R) of positive integers, that satisfy this equation:
(P+2)R+1 - (P+1)Q+1 = 1
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| « Last Edit: Jun 25th, 2008, 8:50am by K Sengupta » |
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TenaliRaman
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I am no special. I am only passionately curious.
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Re: Subtract The Expressions, Get 1
« Reply #1 on: Jun 30th, 2008, 1:11pm » |
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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
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towr
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Re: Subtract The Expressions, Get 1
« Reply #2 on: Jun 30th, 2008, 1:37pm » |
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"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)
linky to sequence A001597
Actually, 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.
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| « Last Edit: Jun 30th, 2008, 1:41pm by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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Eigenray
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Re: Subtract The Expressions, Get 1
« Reply #3 on: Jun 30th, 2008, 7:25pm » |
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Note that we can factor this in more ways if (Q+1) is odd or (R+1) is even. Actually, these both must be the case.
on Jun 30th, 2008, 1:37pm, towr wrote:| Actually, 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. |
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But this is only a special case of Catalan's conjecture (see here for another one). There is an elementary proof.
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| « Last Edit: Jun 30th, 2008, 7:31pm by Eigenray » |
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