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Irrational to Rational (Posted on 2012-06-09) Difficulty: 3 of 5
Determine, with proof, all possible irrational numbers n such that each of n3 - 6n and n4 - 8n2 is a rational number.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Two possible n | Comment 1 of 2
I haven't considered higher order roots but here's what I have so far:

For two rational numbers a, b with √b irrational let
n = a + √b
n = a + b + 2a√b
n = a + 3ab + (3a + b)√b
n^4 = a^4 +6ab + b + (4a + 4ab)√b

n - 6n = a + 3ab -6a  + (3a + b - 6)√b
n^4 - 8n = a^4 + 6ab + b - 8a - b + (4a + 4ab - 16a)√b

For these to be rational the terms in parentheses must both equal zero.  This makes a system:
(3a + b - 6) = 0
(4a + 4ab - 16a) = 0
Solve each for b
b = 6 - 3a
b = 4 - a

So 4 - a = 6 - 3a
a = 1
and b = 3

The only values for  n with the type sought are n = 1 + √3

The rational expressions each come out to 4 or -4


  Posted by Jer on 2012-06-11 12:31:26
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