Difference between revisions of "2006 IMO Problems/Problem 4"
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for <math>x = 1,2</math> no solution. | for <math>x = 1,2</math> no solution. | ||
so assume <math>x > 2</math>. LHS is odd, so writing <math>y = 2n+1</math> gives us <math>2^{x-2}(1+2^{x+1}) = n(n+1)</math>. <math>n,n+1</math> are coprime, and so are <math>2^{x-2}, 1+2^{x+1}</math>. so <math>n = 2^{x-2}, n+1=1+2^{x+1}</math> or vice versa, but both lead to a contradiction | so assume <math>x > 2</math>. LHS is odd, so writing <math>y = 2n+1</math> gives us <math>2^{x-2}(1+2^{x+1}) = n(n+1)</math>. <math>n,n+1</math> are coprime, and so are <math>2^{x-2}, 1+2^{x+1}</math>. so <math>n = 2^{x-2}, n+1=1+2^{x+1}</math> or vice versa, but both lead to a contradiction | ||
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+ | NOTE: This solution does not seem to be correct. Take e.g. <math>x=4</math>. Then <math>LHS=529</math> and <math>y=\pm23</math>. The argument of the solution is promising, but the following is ''not'' true: Let <math>a</math>, <math>b</math> be co-prime and let <math>c</math>, <math>d</math> be also coprime and additionally let <math>ab = cd</math>. Then it must either be <math>(a,b) = (c,d)</math> or <math>(b,a) = (c,d)</math>. |
Revision as of 08:24, 15 August 2013
Problem
Determine all pairs of integers such that
Solution
: LHS integer iff , but then . is a solution. for no solution. so assume . LHS is odd, so writing gives us . are coprime, and so are . so or vice versa, but both lead to a contradiction
NOTE: This solution does not seem to be correct. Take e.g. . Then and . The argument of the solution is promising, but the following is not true: Let , be co-prime and let , be also coprime and additionally let . Then it must either be or .