PELL

1
PELLS EQUATION

• - Nivedita
  

2
Notation

• ??d positive square root of x
• Z ring of integers
• Z?d ab?d a,b in Z

3
Why?

• x2 dy2 0 gt x/y ??d
• If d is not a perfect square, we cant find
  integer solutions to this equation
• The next best thing
• x2 dy2 1 which gives us good rational
  approximations to ??d

4
Approach

• x2 dy21
• Factorizing,
• (xy?d)(x-y?d) 1
• So we look at the ring Z ?d

5
Representation of Zrt(d)

• Trivial
• ab?d ? (a,b)
• Other
• ab?d ?(u,v)
• u a b?d
• v a - b?d

6
Lattice in R x R

• L mx ny m,n in Z
• with x,y two R independent vectors
• x,y is a basis for L
• Fundamental parallelogram FP(L) parallelogram
  formed by x and y
• Z?d in u-v plane is a lattice
• ( basis (1,1) , (??d,-?d) )

7
Observations

• If P ab?d ? (u,v)
• uv a2- db2 (Norm of P)
• Norm is multiplicative
• v Conjugate(P) a b?d
• Note The same definitions of norm, conjugate go
  through for
• P ab?d with a, b in Q

8
Solutions to Pell

• Latticepoints of Z?d with norm 1
• (or)
• Lattice points on hyperbola uv 1

9
Idea and a glitch

• If Norm(P) Norm (Q)
• then Norm (P/Q) 1 !
• But
• P/Q neednt be in Z?d

10

• Any lattice point
• in the shaded
• region has
• absolute value of
• norm lt B

11
Implementation of the idea

• If we can infinitely many lattice points
• inside the region,
• (note, theyll all have norm lt B ),
• then we can find infinitely many
• points which have the same norm r,
• r lt B
• ( norms are integers and finitely many bet
• B and B)

12

• So, identifying lattice points in nice
• sets of R x R seems to be useful.
• Here follows a lemma

13
But whats a nice set in RxR

• Convex
• S is convex if p in S, q in S
• gt line-segment joining p and q is in S
• Centrally symmetric p in S gt -p in S
• Bounded S is bounded if it lies inside a circle
  of radius R for big enough R

14
Minkowskis lemma

• Let L be a lattice in R x R with
• fundamental parallelogram FP.
• If S is a bounded, convex, centrally
• symmetric set such that
• area(S) gt 4 area(FP)
• then S contains a non-zero lattice
• point

15
Infinitely many points with same norm - continued
16

• R(u) the rectangle in the pic satisfies
  minkowski lemma
• conditions for all u gt0 .
• So each R(u) has a non zero lattice point
• No lattice point can be of form (x,0)
• And R(u) becomes narrower as u increases
• So, infinitely many lattice points P with
  norm(P) lt B
• So infinitely many points with the same norm (as
  said before)

17
Go away glitch

• Pick infinitely many points
• Pkakbk?d with same norm r
• Out of this pick infinitely many points such
• that akaj mod r, bkbj mod r for all k, j
• Now evaluate Pk/Pj .
• (by rationalizing denominator)
• It belongs to Z?d

18
Visual proof of Minkowski

• S given set. U p 2p in S

19
Area of U

• Area (U) ¼ Area(S) gt area(FP)
• No. of red squares in U no. of blue squares
  in S

20
Divide U into parallelograms

• Purple lines L(lattice)
• Blue parallelogram FP
• FPa pa p in FP
• Only finitely many a in L
• such that FPa
• intersects U.

21
Translations

• Put Ua (FPa) ? U
• (example red figure)
• Va Ua a
• (the green one)
• So Va lies in FP !
• Area of U
• ? Ua
• ? Va

22

• Area of U gt area of FP
• ? Va gt area of FP
• But all Va lie in FP!
• So some two should overlap
• Va ? Vb is not empty for some a ?b in L
• u a v b ( u, v in U)
• u - c v ( c in L , c b-a ? 0)

23
Voila!

• u- c in U
• c u in U
• (Note U is also convex, centrally symmetric!)
• u in U
• Midpoint of c-u, u in U
• c/2 in U
• c in S