Please answer the below: 6.1 In this exercise, we study equations of the form Dy
ID: 2962810 • Letter: P
Question
Please answer the below:
6.1 In this exercise, we study equations of the form Dy where DEZ 0 is not a square. For some values of D this equation has a solution and for some values of D it doesn't. We do not know how to determine which values of D lead to solvable equations. a) If (zo, yo) is a positive integer solution to Dy show that (ara Dva, 2.royo) is a solution to Pell's equation r2 Dy 1. b) How do you think one could find the solution to Pell's equation from part a)? c) Find a solution (in positive integers) to Aly (Hint: try to find a value of y that makes 41y2 1 a perfect square.)Explanation / Answer
(a)
(x0^2+Dy0^2)^2 - D(2x0y0)^2
=
x0^4 +D^2y0^4+2Dx0^2y0^2 - 4Dx0^2y0^2
=
(x0^2-Dy0^2)^2
=
(-1)^2
= 1
thus proved
(c)
(32, 5) is a solution