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Home » Olympiad Section » Number Theory » Number Theory Open Questions
 
Integers
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Zaq
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#1
Integers
Show that if m, n and r are positive integers and: 1 + m + n\sqrt {3} = (2 + \sqrt {3})^{2r - 1 then m is a perfect square.

PostPosted: Thu Oct 22, 2009 2:45 am  Back to top 
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#2
(2+\sqrt 3)^a=m+1+n\sqrt 3, (2-\sqrt 3)^a=m+1-n\sqrt 3\to 1=(m+1)^2-3n^2.
Therefore m(m+2)=3n^2 and if (6,m)=1, then 3\not m, (m,m+2)=1 . so m is square,
m=2^a-1\mod 3=1\mod 3 because a is odd and m+1=0\mod 2 because a is odd. It give result.

PostPosted: Thu Oct 22, 2009 8:05 am  Back to top 
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