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Topic: An Astute Integer Triplet Puzzle (Read 1823 times) 

K Sengupta
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An Astute Integer Triplet Puzzle
« on: Mar 11^{th}, 2007, 5:19am » 
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Analytically determine all possible integer triplets (x ,y, z) satisfying this equation: sqrt(x+ sqrt(x+ sqrt(x + sqrt(x ......+ sqrt(x + sqrt(x + y))))) = z; with x < 100,000 and y< 100,000; where: n = n for n>=0 and; = n for n<0. The square root symbol in the above relationship is repeated 2007 times.

« Last Edit: Mar 11^{th}, 2007, 5:31am by K Sengupta » 
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Barukh
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Re: An Astute Integer Triplet Puzzle
« Reply #1 on: Mar 14^{th}, 2007, 12:24am » 
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It depends on the interpretation of the sqrt... In its positive sense, my guess is that (a^{2}  a, a, a) is the only family of solutions.


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Aryabhatta
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Re: An Astute Integer Triplet Puzzle
« Reply #2 on: Mar 14^{th}, 2007, 12:01pm » 
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I guess we do an induction on the number of times the square root repeats and show that the solutions described by Barukh are the only ones... For an n square root equation, It would be easy to prove that sqrt(x+y) is also an integer and then replace sqrt(x+y) with u and solve the same equation with with n1 square roots with y replaced by u. btw, why do you insist on using adjectives like "Astute", "Ingenious" etc ?

« Last Edit: Mar 14^{th}, 2007, 12:07pm by Aryabhatta » 
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Barukh
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Re: An Astute Integer Triplet Puzzle
« Reply #3 on: Mar 14^{th}, 2007, 11:00pm » 
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on Mar 14^{th}, 2007, 12:01pm, Aryabhatta wrote:btw, why do you insist on using adjectives like "Astute", "Ingenious" etc ? 
 Maybe, because this makes the problems more Aesthetic?


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ThudnBlunder
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Re: An Astute Integer Triplet Puzzle
« Reply #4 on: Mar 15^{th}, 2007, 12:31am » 
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on Mar 14^{th}, 2007, 12:01pm, Aryabhatta wrote: btw, why do you insist on using adjectives like "Astute", "Ingenious" etc ? 
 Besides the fact that the number of letters in each of the words 'magic', 'astute', 'unusual', 'greatest', 'ingenious', and 'analytical' are in arithmetic progression, perhaps because an appropriate juxtaposition of judiciously chosen adjectives confers less ambiguity upon the expressions constituting the thread titles, thus reducing the probability of anomalous interpretations of the problems under reference.

« Last Edit: Mar 26^{th}, 2007, 11:33am by ThudnBlunder » 
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.



Aryabhatta
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Re: An Astute Integer Triplet Puzzle
« Reply #5 on: Mar 15^{th}, 2007, 12:52pm » 
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on Mar 15^{th}, 2007, 12:31am, ThudanBlunder wrote: Besides the fact that the number of letters in the words 'astute', 'integer', 'greatest', 'ingenious', and 'trivariate' are in arithmetic progression, perhaps because a suitable juxtaposition of appropriate adjectives bestows less ambiguity on the expressions constituting the thread titles, thus reducing the probability of anomalous interpretations of the problems under reference. 
 Lol! I guess that must be it. Please take no offense KSengupta. There is no such intention.

« Last Edit: Mar 15^{th}, 2007, 1:11pm by Aryabhatta » 
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ThudnBlunder
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Re: An Astute Integer Triplet Puzzle
« Reply #6 on: Mar 15^{th}, 2007, 6:09pm » 
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on Mar 15^{th}, 2007, 12:52pm, Aryabhatta wrote: Please take no offense KSengupta. There is no such intention. 
 Of course not, KS.


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K Sengupta
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Re: An Astute Integer Triplet Puzzle
« Reply #7 on: Mar 22^{nd}, 2007, 8:07am » 
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on Mar 15^{th}, 2007, 12:31am, ThudanBlunder wrote: Besides the fact that the number of letters in each of the words 'magic', 'astute', 'unusual', 'greatest', 'ingenious', and 'analytical' are in arithmetic progression, perhaps because an appropriate juxtaposition of suitable adjectives confers less ambiguity upon the expressions constituting the thread titles, thus reducing the probability of anomalous interpretations of the problems under reference. :D 
 Thanks for the vote of confidence TB. I may add that, the said methodology also facilitates the proceedure corresponding to reduction of the probability of recrudescence of a multiplicity of threads with the same title, but possessing different concepts, which I understand is at direct variance with the provisions governing this website.

« Last Edit: Mar 22^{nd}, 2007, 8:18am by K Sengupta » 
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