Patterns Occurring during GEMTEX Confrac Expansion P.L.Douillet - - PowerPoint PPT Presentation

GEMTEX P.L.Douillet 8/01/2004

E.N.S.A.I.T. (Roubaix)

Patterns Occurring during Confrac Expansion

• f Quadratic Numbers

Pierre L. Douillet douillet@ensait.fr

GEMTEX P.L.Douillet 8/01/2004

x0 1 qj xj ; xj1 1÷ xjqj x0 q0 1 q1 1 1 qj1/xj

Continued Fractions

key idea : start from a positive number subtract as much as you can invert ; repeat accelerating :

• btaining :

GEMTEX P.L.Douillet 8/01/2004

x0 j xj qj 3 1 7 1 151/x3 333x322 106x37

Let's have an example

starting from 3.1415927, we obtain 3.1415927 3 1 7.0625100 7 2 15.9971868 15 so that

GEMTEX P.L.Douillet 8/01/2004

P1 Q1 22 7 , P2 Q2 333 106 P20, P11, Pjqj Pj1Pj2 Q20, Q11, Qjqj Qj1Qj2

Stepwise computation of convergents

example continued recurrence formula

GEMTEX P.L.Douillet 8/01/2004

• Pj1

Qj1 Pj Qj

• 1

Qj Qj1

N/Dx0 < 1/D 2 N/D

Best approximation property

fundamental property moreover, implies is a convergent

GEMTEX P.L.Douillet 8/01/2004

2.4545 < x < 2.4546 x 2.4545 2, 2, 4, 1, 181, 2.4546 2, 2, 5, 151, 3, x 2, 2, 4, 1 2, 2, 5 27 11

extracting the exact value from an approximation

suppose that where is known to have a small denominator compute conclude

GEMTEX P.L.Douillet 8/01/2004

vnRH 1 22 1 n 2

examples of appliability

Balmer , Galton, etc gears dimensioning factoring great integers

GEMTEX P.L.Douillet 8/01/2004

• x

Prob ank log2 1 1 k1 2 1 n

n 1

qk

n

• n

1

qk 2.685452...

further properties

finite expansion rational periodic expansion quadratic for quite all , and therefore (Khintchine) while

GEMTEX P.L.Douillet 8/01/2004

z 2bzc0

n D m D

2 m D1 mod4 6 45 12;1,2,2,2,1

quadratic integers

a quadratic integer is a root of a monic integer polynomial i.e. first kind, generated by second kind, generated by where is odd and palindromic pattern:

GEMTEX P.L.Douillet 8/01/2004

4276 1828485

• 8552;12,16,12

1 2 4275 1828485

4275;1,1,5,1,1,7,1,1,5,1,1 12 153

• 24;2,1,2,2,2,1,2

1 2 11 153

11;1,2,5,1,5,2,1 37 1397

• 74,2,1,1,1,10,18,1,1,2,6,...

...2,1,1,18,10,1,1,1,2

1 2 37 1397

37;5,3,5

from to and conversely

GEMTEX P.L.Douillet 8/01/2004

ω θ

173 217

period of versus period of

GEMTEX P.L.Douillet 8/01/2004

h z az b cz d ; adbc 0 h z 1 prop_to a b c d z 1 h z q 1 1 0 z

fractional linear transforms

definition : matrix representation

• ne confrac step is

GEMTEX P.L.Douillet 8/01/2004

z 1 eigenvect_of

• L

1

qi 1 1 0 M q 1 1 0 . u v v w u z 2 quz vq w 0 q u v wdet q 2u 2 4 vq w

patterns

periodicity: palindromy: quadratic integer : therefore

GEMTEX P.L.Douillet 8/01/2004

u mod4 ; v mod2 ; w mod4

• nly 12 signatures

signature = mid det u v w D mod8 D mod4

• 1

1 5 1 ,2 2v1 no + 1 1 2 5 1 ,2 2 1 1 1,5 1

0, 1 ,2,3 1+2v1

even

• 1

2

0, 1 ,2,3

2 1

0,2

±1 1 5

0,3

• dd
• ±1

1 5

0,3

1 ±1 1,5

GEMTEX P.L.Douillet 8/01/2004

D 1 mod4 length 3×length

• theorem 1

for all , not a perfect square, equality occurs if, and only if, the pattern is "any, one, one" (with middle element odd or none) (cf exemple 1, slide )

GEMTEX P.L.Douillet 8/01/2004

D 1 mod4 length 5×length

• 3

D 5 mod8 q 2q q/2 q/2 1,1

theorem 2

for all , not a perfect square, equality occurs if, and only if, all quotients

• ccuring in the pattern are

and u is odd in such a case , each lead to three numbers

• r

for each a sequence appear (cf exemple 3, slide )

GEMTEX P.L.Douillet 8/01/2004

ω θ

173 217

conclusion

GEMTEX P.L.Douillet 8/01/2004

4 , 4 17 4 4 1 1, 0, 1 4;4,4 2 5 21;4,4 1

2 21 461

38;4,4 19 370 55;4,4 1

2 55 3077

72;4,4 36 1313 1313 1

2 35 1313 1;1,1,1,1,1,1,1,1

D m;1,1, .. ,1,1

3k2

D 2n;4, .. ,4

k

nothing but fours

pattern matrix sign

• when