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For which of the following values of k does have a solution with x, y integers:
(a) (b) (c) (d)
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Recall Fermat's Theorem on the Sums of Two Squares: A number n is a sum of two squares if and only if all prime factors of the form have even exponent in the prime factorization of n.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
(a) Since , Fermat's theorem says that in case (a) we must have a solution. (The curiously minded person should note that as then multiplying both sides by gives ).
(b) As and , this case has no solutions.
(c) As and , this case has no solutions.
(d) As and , this case has no solutions.