pell equations

Question 1

pell equation:

Answer 1

an equation of the form x^2-dy^2=1, where d is a positive integer
we only care about integer solutions
the trivial one is x=1, y=0
we care only for positive integer solutions cause we are squaring them so can take all combos of negative and positive x and y

Question 2

if d is a square number, the pell equation has:

Answer 2

only the trivial solution so mostly we won’t look at those, only non-square d

Question 3

Z[root(d)]:

Answer 3

{a+broot(d): a,b integers} - we factorise a pell equation as x+-yroot(d) hence this stuff

Question 4

norm map:

Answer 4

N:Z[root(d)]->Z (integers) by N(a)=aa, where a is the conjugate of a
so N(c)=N(a+broot(d))=a^2+db^2=N(a-broot(d))=N(c*)

Question 5

for two sets of positive solutions to a pell equation:

Answer 5

x1<x2 <=> y1<y2
x1=x2 <=> y1=y2

Question 6

if a pell equation has a positive solution (x1,y1):

Answer 6

then it has infinite distinct positive solutions of the form (xn,yn) where xn+ynroot(d)=(x1+y1root(d))^n
in fact if it has a non-trivial solution, then it has a minimal positive solution (x1,y1) and All other solutions have the above form or a different combo of negatives

Question 7

fundamental solution:

Answer 7

that minimal positive solution (x1,y1)

Question 8

|cqn-pn|<=:

Answer 8

|cb-a|, where c is the irrational simple continued fraction pn and qn are based on, a,b integers, and 1<=b<q(n+1)

Question 9

if |(a/b)-c|<1/2b^2:

Answer 9

then a/b is a convergent of c, c irrational, b>0

Question 10

if 0<x^2-dy^2<root(d):

Answer 10

then x/y is a convergent of root(d)

Question 11

root(d) as a continued fraction expansion:

Answer 11

d not a perfect square, root(d)=[x0: x1,…,xr] (repeat bar over x1,…,xr)
the positive int solutions to x^2-dy^2=1 are the numerators and denominators x=p(kr-1) and y=q(kr-1) where k is any positive integer that such that kr even
the fundamental solution is (pn,qn), n=2r-1 if r is odd and r-1 if r is even