Question

34.1. In this chapter we have shown that Pell's equation x2- Dy2-1 always has a so- lution in positive integers. This exercise explores what happens if the 1 on the right-hand side is replaced by some other number (a) For each 2 < D < 15 that is not a perfect square, determine whether or not the equation x2 - Dy2 - -1 has a solution in positive integers. Can you determinea pattern that lets you predict for which D's it has a solution? (b) If (zo, yo) is a solution to 22 - Dy2 - -1 in positive integers, show that (rj -i Dyj,2ro/o) is a solution to Pell's equation z2 - Dy2-1. (c) Find a solution to x2-41уг--1 by plugging in y 1, 2, 3,.. . until you find a value for which 41y2 - 1 is a perfect square. (You won't need to go very far.) Use your answer and (b) to find a solution to Pell's equation x2 - 41y2- 1 in positive ntegers (d) If (xo, yo) is a solution to the equation r2- Dy2-M, and if (xı, yı) is a solution to Pell's equation 2 - Dy2-1, show that (o1 + Dyoyi,2oyi yoı) is alsoa solution to the equation x2 - Dy2 - M. Use this to find five different solutions in positive integers to the equation x2 - 2y2-7.

Answer 1

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