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   An Astute Integer Triplet Puzzle
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   Author  Topic: An Astute Integer Triplet Puzzle  (Read 1700 times)
K Sengupta
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An Astute Integer Triplet Puzzle  
« on: Mar 11th, 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 11th, 2007, 5:31am by K Sengupta » IP Logged
Barukh
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Re: An Astute Integer Triplet Puzzle  
« Reply #1 on: Mar 14th, 2007, 12:24am »
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It depends on the interpretation of the sqrt...
 
In its positive sense, my guess is that (a2 - 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 14th, 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 n-1 square roots with y replaced by u.
 
btw, why do you insist on using adjectives like "Astute", "Ingenious" etc ?
« Last Edit: Mar 14th, 2007, 12:07pm by Aryabhatta » IP Logged
Barukh
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Re: An Astute Integer Triplet Puzzle  
« Reply #3 on: Mar 14th, 2007, 11:00pm »
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on Mar 14th, 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?  Wink
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Re: An Astute Integer Triplet Puzzle  
« Reply #4 on: Mar 15th, 2007, 12:31am »
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on Mar 14th, 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.
 
  Cheesy
« Last Edit: Mar 26th, 2007, 11:33am by ThudnBlunder » IP Logged

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Re: An Astute Integer Triplet Puzzle  
« Reply #5 on: Mar 15th, 2007, 12:52pm »
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on Mar 15th, 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.
 
Cheesy
 
 

 
Lol!
I guess that must be it.  Tongue
 
 
Please take no offense KSengupta. There is no such intention.
« Last Edit: Mar 15th, 2007, 1:11pm by Aryabhatta » IP Logged
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Re: An Astute Integer Triplet Puzzle  
« Reply #6 on: Mar 15th, 2007, 6:09pm »
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on Mar 15th, 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 22nd, 2007, 8:07am »
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on Mar 15th, 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 22nd, 2007, 8:18am by K Sengupta » IP Logged
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