Question 563151: Let A, B, C, be integers such that  is an integer. Prove that  ,  ,  are integers.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! You can let  where k is an integer, then  . Cubing both sides,
Here, we can take this equation "modulo 1" by eliminating all the integer expressions (b,c,k^3,3ka).
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Factor LHS
 \equiv -3k^2\sqrt{a} - a\sqrt{a} (mod 1)) Note that you can replace  with  . Modulo 1, this is equivalent to -sqrt(a).
 \equiv -3k^2\sqrt{a} - a\sqrt{a} (mod 1))
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Here, we show that ) i.e. it is an integer. I'll let you finish the proof that sqrt{a}, sqrt[3]{b}, and sqrt[3]{c} have to be integers. Pretty daunting problem...unfortunately we cannot assume the converse of the statement is true (i.e. if ... are integers then ... is an integer).
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