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Surprisingly, I managed to prove the following result:
For any rational , there exists an irrational such that is rational.

Suppose . consider the equation . We aim at proving that there is no rational satisfies this equation.

Assume in contrary that such an exists.
(1) : This gives , i.e., , violating our assumption.

(2) : Write for some positive coprime integers and .

Since is rational, is also rational. Therefore for some positive coprime integers and with .
Moreover, implies . So .
Substituting these formulae into the original equation, we get , or .
Since is irreducible and is also irreducible, we have and .
For , i.e., , since , we have or .
For , we have ; when , we have , which has or as its integer solutions.
(i) . Since , such does not exist. There is no solution.
(ii) . Put into the equation, and get , which does not have integer solution.
(iii) , . We have .
Try : , which is impossible. Try : is, again, impossible. So there is no solution either.
In conclusion, there is no rational satisfies the equation.

(3) : Write , where and are positive coprime integers. Again, for the same reason.

Plug in all these substitution to the equation: , or .
Again, by irreducibility, we have and .
(i) . Then and .
Note that . So . So , which is impossible.
(ii) . Then . Hence , which is impossible.
In conclusion, there is no rational number satisfies the original equation.

Now suppose . Consider . If there exists a rational satisfying this equation, then satisfies , contradicts to the above conclusion.

Now we prove that, for any rational , is rational for some irrational .

For , since , there exists with . It is proved that cannot be rational. So must be irrational.
For , we have . So there exists with . Again, cannot be rational by previous arguments. So must be irrational.
For , it is trivial that , for which is irrational but is rational.

This ‘proof’ is too clumsy and perhaps there are lots of errors. Please help me check if the arguments are valid.


Therefore do not worry about tomorrow, for tomorrow will worry about itself. Each day has enough trouble of its own.