Question

9.14 Theorem. f the natural mumber N is a perfect square, then the Pell equation Ny 1 has no non-trivial integer solutions. After all this talk about trivial solutions, let's at least confirm that in some cascs non-trivial solutions do cxist. 9.15 Exercise. Find, by trial and error at least two non-trivial solutions to each of the Pell equations x2-2y2 I and x-3y21 Rolstcred by the cxistence of solutions for N 2 and N 3, our focus from this point forward will be on finding non-trivial solutions to the Pell equations x- Ny where N is a natural number that is not a perfect amnbs New solutions from old For a positive integer N that is not a perfect square, the non-trivial solutions to x-Ny come to us in natural groups of four since the square of a negative number is positive. 9.16 Question. To know all the integer solutions to a Pell equation, why does it suffice to know just the positive integer solutions? One solution to a Pell cquation gives rise to related ones hy taking nega- tives, but there are other ways to lake some solutions and combine them to create other solutions. Since I times I cquals 1, multiplication of solutions also gives a new solution. Here is what we mean. 9.17 Theorem. Suppase N is a natural mumber and the Pell equation x2-Ny1 has two solutions, namely, a2-Nh?= l and e-Na2 = for some integers a, b, c, and d. Then xac Nbd and y ad +be is also an integer solution to the Pell equation x Ny . That is, (aeNhd)2 N(ad + be-

Answer 1

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