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Show that if d is an odd squarefree integer then a necessary condition for the equation to have an integer solution is that all the primes dividing d must be of the form .
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.
Your first reaction might be to use the theory of Pell equations and continued fractions on this problem, but there is a much more elegant solution.
Suppose that in fact d was of the form . What can we conclude about prime divisors of d?
Recall that the product of two primes of the form must be a number of the form .
Use a sub-case of quadratic reciprocity or Euler's criterion to finish the problem.
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.
We proceed as in the hints. Suppose that in fact d was of the form . Then since
we know that there must be a prime p of the form that divides d. Reducing modulo p gives
This means that . However, Euler's criterion tells us that and this is a contradiction.