4) Pell equations

Question 1

What are pell equations

Answer 1

Question 2

What is the set Z[d] for a non-square positive integer d

Answer 2

Question 3

What is the norm map for elements of Z[d], and how is it calculated

Answer 3

Question 4

What is the relationship between solutions to Pell’s equation and elements of Z[d] with norm 1

Answer 4

Question 5

What property does the norm map satisfy for elements α,β ∈ Z[d]

Answer 5

Question 6

If x^2 −dy^2 = 1 has one positive solution, how can we generate infinitely many positive solutions

Answer 6

Question 7

What is the fundamental solution to Pell’s equation
x^2 − dy^2 = 1

Answer 7

Question 8

For two positive solutions (x1,y1) and (x2,y2) of x^2 − dy^2 =1, how are the solutions ordered

Answer 8

Question 9

If Pell’s equation x^2 −dy^2 = 1 has non-trivial solutions, how can we describe all positive integer solutions

Answer 9

Question 10

What does the lemma say about the quality of approximation given by the convergents of an irrational number α

Answer 10

Question 11

How can we recognise whether a fraction a/b is a convergent of an irrational number α

Answer 11

Question 12

If d is a positive integer that is not a perfect square, and (x,y) are positive integers such that 0 < x^2 − dy^2 < d , what can we say about x/y

Answer 12

Question 13

For the convergents pn/qn of √d , when is (pn,qn) a solution to Pell’s equation x^2 − dy^2 = 1

Answer 13

Question 14

For a positive integer d that is not a perfect square, what is the fundamental solution to x^2 − dy^2 = 1 in terms of the continued fraction expansion of √d

Answer 14