Science:Math Exam Resources/Courses/MATH437/December 2006/Question 06
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Question 06 

Let , where is an integer. Show that has no integer solutions. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

This time, your first reaction might be to use the theory of Pell equations and continued fractions on this problem will steer you in the correct direction. 
Hint 2 

Recall from the theory of Pell equations that has a solution if and only if the period of is even. 
Hint 3 

Computing the period can be done abstractly by hand. 
Hint 4 

As a final learning moment, it should be noted that the sufficiency in the previous problem is not true and this problem gives a counter example when say 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Proceeding as in the hints, we see it suffices to show that the period of is even. First notice that
and thus in the continued fraction expansion. As
we see that after cross multiplying that
As
cross multiplying gives us that
and thus as we know ,
Here we reduce back to the case of . Hence, the continued fraction expansion is given by . Thus the period of the continued fraction expansion is even and the theory of Pell equations states that the equation has no solution. 