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K Sengupta
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An All Possible Value Puzzle
« on: Nov 3rd, 2006, 7:05am » |
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A, B and C are positive integers.
Determine all possible values of C, for which the equation:
2^A – 5^B = C , possesses exactly two distinct solutions.
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| « Last Edit: Nov 3rd, 2006, 7:06am by K Sengupta » |
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ThudnBlunder
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Re: An All Possible Value Puzzle
« Reply #1 on: Nov 3rd, 2006, 12:07pm » |
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Nice problem!
So (A1, B1) and (A2, B1) are not distinct?
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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Icarus
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Re: An All Possible Value Puzzle
« Reply #2 on: Nov 3rd, 2006, 3:59pm » |
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The question doesn't come up, for if the values of B and C are the same, the values of A must be as well.
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When their digits are reversed? " - Anonymous
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ThudnBlunder
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Re: An All Possible Value Puzzle
« Reply #3 on: Nov 3rd, 2006, 4:27pm » |
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on Nov 3rd, 2006, 3:59pm, Icarus wrote:| The question doesn't come up, for if the values of B and C are the same, the values of A must be as well. |
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I understand now. C is constant (a positive integer) and we need to find what values of this constant allow exactly 2 distinct solutions.
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| « Last Edit: Nov 3rd, 2006, 4:37pm by ThudnBlunder » |
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Barukh
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Re: An All Possible Value Puzzle
« Reply #4 on: Nov 3rd, 2006, 5:59pm » |
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One solution (to start with): C = 3.
Are there any others? 
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Eigenray
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Re: An All Possible Value Puzzle
« Reply #5 on: Nov 4th, 2006, 12:27am » |
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Hmmm...
2x(24r-1) = 5y(52s-1).
We'd like to conclude r=s=1.
Suppose, for example, that 2|r. Then 17|24r-1, so 17|52s-1, so 8|s. Then 11489|52s-1, so 11489|24r-1, so 1436|r. Etc.?
The number 3 might be important. The number of times 3 divides r is the same as the number of times 3 divides s, for example. We also know the number of times 5 divides r is y-1, and the number of times 2 divides s is x-3, but I don't know if that helps.
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Sameer
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Re: An All Possible Value Puzzle
« Reply #6 on: Dec 18th, 2006, 9:14am » |
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Actually thanks for T&B's link in another post. I think this riddle just got buried. So this way it will be bumped!! 
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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Eigenray
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Re: An All Possible Value Puzzle
« Reply #7 on: Dec 18th, 2006, 11:14pm » |
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Well, 2x(24r-1) = 5y(52s-1) has only one solution with 0<r,s<106 (much past that I run out of memory cause I'm too lazy to write an efficient program). But I haven't made any real progress.
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Sameer
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Re: An All Possible Value Puzzle
« Reply #8 on: Dec 19th, 2006, 10:41am » |
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anyway to do this analytically? 
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"Obvious" is the most dangerous word in mathematics.
--Bell, Eric Temple
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington, quoted in Bridges to Infinity
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Eigenray
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Re: An All Possible Value Puzzle
« Reply #9 on: Jun 23rd, 2007, 8:10pm » |
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K Sengupta, did you have a solution?
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K Sengupta
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Re: An All Possible Value Puzzle
« Reply #10 on: Jun 26th, 2007, 1:49am » |
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on Jun 23rd, 2007, 8:10pm, Eigenray wrote:| K Sengupta, did you have a solution? |
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I truly do not have the solution to this one.
Prima facie, this problem occurred to my mind having it's unmistakable similarity with the famous 2^x - 3^y = 7 problem which possesses an elementary solution.
Having regard to its simplicity, I naively assumed that the 2^x - 5^y case would also lend itself to a similar treatment, a premise which has since proved erroneous.
In conclusion, I would like to thank you for your brilliant treatment of the various facets corresponding to the problem under reference, with a hope that someday, an analytic solution to the foregoing problem could be found.
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