Science:Math Exam Resources/Courses/MATH437/December 2006/Question 05

MATH437 December 2006

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Question 05
Show that if d is an odd squarefree integer then a necessary condition for the equation $\displaystyle x^{2}-dy^{2}=-1$ to have an integer solution is that all the primes dividing d must be of the form $\displaystyle 4n+1$.

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
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.

Hint 2
Suppose that in fact d was of the form $\displaystyle 4n+3$. What can we conclude about prime divisors of d?

Hint 3
Recall that the product of two primes of the form $\displaystyle 4k+1$ must be a number of the form $\displaystyle 4m+1$.

Hint 4
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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We proceed as in the hints. Suppose that in fact d was of the form $\displaystyle 4n+3$. Then since $\displaystyle (4k+1)(4m+1)=4(4km+k+m)+1$ we know that there must be a prime p of the form $\displaystyle p=4j+3$ that divides d. Reducing modulo p gives $\displaystyle x^{2}\equiv -1\mod {p}$ This means that $\displaystyle \left({\frac {-1}{p}}\right)=1$. However, Euler's criterion tells us that $\displaystyle 1=\left({\frac {-1}{p}}\right)\equiv (-1)^{(p-1)/2}=(-1)^{2j+1}=-1\mod {p}$ and this is a contradiction.

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