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

E.N.S.A.I.T. (Roubaix) Patterns Occurring during GEMTEX Confrac Expansion P.L.Douillet of Quadratic Numbers 8/01/2004 Pierre L. Douillet douillet@ensait.fr

� key idea : x 0 start from a positive number subtract as much as you can 1 invert ; repeat q j � x j ; x j � 1 � 1÷ x j � q j � accelerating : GEMTEX P.L.Douillet 1 x 0 � q 0 � � obtaining : 1 8/01/2004 q 1 � 1 � �� � q j � 1/ x j Continued Fractions

x 0 � � starting from 3.1415927, we obtain j x j q j 0 3.1415927 3 1 7.0625100 7 GEMTEX 2 15.9971868 15 P.L.Douillet � 333 x 3 � 22 8/01/2004 1 � so that � � 3 � 106 x 3 � 7 1 7 � 15 � 1/ x 3 Let's have an example

� example continued P 1 , P 2 � 22 � 333 � � Q 1 Q 2 7 106 � recurrence formula GEMTEX P � 2 � 0, P � 1 � 1, P j � q j P j � 1 � P j � 2 P.L.Douillet Q � 2 � 0, Q � 1 � 1, Q j � q j Q j � 1 � Q j � 2 8/01/2004 Stepwise computation of convergents

� fundamental property � ��� ���� P j � 1 � P j 1 � Q j � 1 Q j Q j Q j � 1 GEMTEX P.L.Douillet N / D � x 0 < 1/ D 2 N / D � moreover, implies 8/01/2004 is a convergent Best approximation property

2.4545 < x < 2.4546 x � suppose that where is known to have a small denominator 2.4545 � 2, 2, 4, 1, 181, ��� � compute 2.4546 � 2, 2, 5, 151, 3, ��� GEMTEX P.L.Douillet x � 2, 2, 4, 1 � 2, 2, 5 � 27 � conclude 8/01/2004 11 extracting the exact value from an approximation

2 2 � 1 1 v n � R H � Balmer , Galton, etc n 2 � gears � dimensioning GEMTEX � factoring great integers P.L.Douillet 8/01/2004 examples of appliability

� finite expansion rational � � periodic expansion quadratic � x � for quite all , 1 Prob a n � k GEMTEX � � log 2 1 � k � 1 2 P.L.Douillet � and therefore (Khintchine) 8/01/2004 n n n n � � 1 q k � � q k � 2.685452... while 1 1 further properties

� a quadratic integer is a root of a monic z 2 � bz � c � 0 integer polynomial i.e. � � n � D � first kind, generated by � � m � D GEMTEX � second kind, generated by 2 m D � 1 mod4 P.L.Douillet where is odd and 8/01/2004 � palindromic pattern: 6 � 45 � 12;1,2,2,2,1 quadratic integers

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 � GEMTEX 11;1,2,5,1,5,2,1 P.L.Douillet 74,2,1,1,1,10,18,1,1,2,6,... 37 � 1397 � ...2,1,1,18,10,1,1,1,2 8/01/2004 1 2 37 � 1397 � 37;5,3,5 from � to � and conversely

217 GEMTEX ω P.L.Douillet 8/01/2004 θ 173 period of � versus period of �

� definition : h z � az � b ; ad � bc � 0 cz � d � matrix representation a b h z z GEMTEX prop _ to 1 c d 1 P.L.Douillet 8/01/2004 q 1 h z � z � one confrac step is 1 0 fractional linear transforms

q i 1 L � z eigenvect _ of � periodicity: 1 1 0 1 q 1 u v M � � palindromy: . v w 1 0 GEMTEX u z 2 � quz � vq � w � 0 P.L.Douillet � quadratic integer : 8/01/2004 q � � u � v w det � therefore � � q 2 u 2 � 4 vq � w patterns

u mod4 ; v mod2 ; w mod4 � signature = mid det u v w D mod8 D mod4 � � � 1 0 1 5 1 ,2 2v 1 no + 1 1 2 5 1 ,2 0 2 1 1 1,5 GEMTEX 0 1 0 0, 1 ,2,3 1+2v 1 P.L.Douillet even - 0 1 2 0, 1 ,2,3 0 0,2 2 1 0 8/01/2004 0,3 ±1 0 � 1 5 0,3 odd - ±1 1 0 5 0 1 ±1 1,5 only 12 signatures

D � 1 mod4 � for all , not a perfect square, length � � 3 ×length � � equality occurs if, and only if, the pattern � is "any, one, one" GEMTEX (with middle element odd or none) P.L.Douillet (cf exemple 1, slide ) 8/01/2004 theorem 1

D � 1 mod4 � for all , not a perfect square, length � � 5 ×length � � equality occurs if, and only if, all quotients occuring in the pattern are � 3 and u is odd GEMTEX � D � 5 mod8 P.L.Douillet � in such a case , q /2 q 2 q each lead to three numbers or 8/01/2004 q /2 for each a sequence 1,1 appear (cf exemple 3, slide ) theorem 2

217 GEMTEX ω P.L.Douillet 8/01/2004 θ 173 conclusion

17 4 � pattern 4 , 4 matrix sign 1, 0, 1 4 1 � 1 4;4,4 21;4,4 � 2 � 5 2 21 � 461 � 1 38;4,4 55;4,4 � 19 � 370 2 55 � 3077 GEMTEX 72;4,4 � 36 � 1313 P.L.Douillet 8/01/2004 � 1313 � 1 2 35 � 1313 � 1;1,1,1,1,1,1,1,1 � � D � m ;1,1, .. ,1,1 � D � 2 n ;4, .. ,4 when � 3 k � 2 k nothing but fours