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   Author  Topic: Integral Solutions  (Read 1322 times)
l4z3r
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Integral Solutions  
« on: Aug 29th, 2008, 4:46am »
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a function is defined as:
 
 
  f: Z(+) --> Z
 
  f(m,n) = (n3 + 1)/ (mn - 1)
 
 
where Z(+) denotes the set of positive integers and Z the set of integers.
 
Find all the solutions for (m,n)
 
EDIT: f(m,n) not f(x)
« Last Edit: Aug 29th, 2008, 5:42am by l4z3r » IP Logged
towr
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Re: Integral Solutions  
« Reply #1 on: Aug 29th, 2008, 5:23am »
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Shouldn't the x in f(x) come into it somewhere?
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l4z3r
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Re: Integral Solutions  
« Reply #2 on: Aug 29th, 2008, 5:42am »
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ah. meant f(m,n). sorry.
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SMQ
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Re: Integral Solutions  
« Reply #3 on: Aug 29th, 2008, 5:58am »
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So, in other words, "find all , such that ( + 1) / ( - 1) ", right?
 
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l4z3r
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Re: Integral Solutions  
« Reply #4 on: Aug 29th, 2008, 7:00am »
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yes. hint:use n3+1   1(mod3) and mn-1   -1 (mod n) (number theory)
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Re: Integral Solutions  
« Reply #5 on: Aug 29th, 2008, 11:53am »
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If (m,n) is a solution, then (m, (m2+n)/(mn-1)) is also; then use the ideas that appear here (and which should appear here).
 
Or, you can proceed more directly by writing n3+1 = (mn-1)((an-m)n-1) and bounding.
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l4z3r
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Re: Integral Solutions  
« Reply #6 on: Aug 30th, 2008, 5:34am »
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hmm. good one. I agree with the first part.
 
If (m,n) is a solution, then (m, (m2+n)/(mn-1)) is also
 
but, instead of (mn-1)((an-m)n-1) i feel a better alternative would be (kn-1)(mn-1). Gives the answer in lesser steps, i think.
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