Some applications of Number Theory. Number Theory in Science and Communication

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December 13, 2005 Nehru Science Center, Mumbai Mathematics in the real life: The Fibonacci Sequence and the Golden Number Michel Waldschmidt Université P. et M.

1 December 13, 2005 Nehru Science Center, Mumbai Mathematics in the real life: The Fibonacci Sequence and the Golden Number Michel Waldschmidt Université P. et M. Curie (Paris VI) Alliance Française Manfred R. SCHROEDER Number Theory in Science and Communication With application in Cryptography, Physics, Digital Information, Computing and Self-Similarity Some applications of Number Theory Cryptography, security of computer systems Data transmission, error correcting codes Interface with theoretical physics Musical scales Numbers in nature Springer Series in Information Sciences 1985

2 How many ancesters do we have? Bees genealogy Sequence: 1, 2, 4, 8, 16 E n+1 =2E n E n =2 n Number of females at level n+1 = Bees genealogy total population at level n Number of males at level n+1 = number of females at level n = = = = 2 Sequence: 1, 1, 2, 3, 5, 8, F n+1 =F n + F n-1 Fibonacci (Leonardo di Pisa) Pisa!1175,!1250 Liber Abaci! 1202 F 0 =0, F 1 =1, F 2 =1, F 3 =2, F 4 =3, F 5 =5, = = 1

3 Modelization of a population Adult pairs Young pairs First year Second year Theory of stable populations (Alfred Lotka) Assume each pair generates a new pair the first two years only. Then the number of pairs who are born each year again follow the Fibonacci rule. Third year Fourth year Fifth year Sixth Year Arctic trees In cold countries, each branch of some trees gives rise to another one after the second year of existence only. Sequence: 1, 1, 2, 3, 5, 8, F n+1 = F n + F n-1 First year Exponential Sequence Representation of a number as a sum of distinct powers of 2 Second year Third year Fourth year 51=32+19, 32=2 5 19=16+3, 16=2 4 3=2+1, 2=2 1, 1=2 0 51= Binary expansion Number of pairs: 1, 2, 4, 8, E n =2 n

4 Decimal expansion of an integer Representation of an integer as a sum of Fibonacci numbers 51 = 5! = 20! N a positive integer F n the largest Fibonacci number " N Hence N= F n + remainder which is < F n-1 Repeat with the remainder 31 F 42 =3 =1 F 53 =5 =2 Example F 64 1 =8 =3 =1 F 9 =34 51= F F 75 2 =13 =5 F 86 3 =21 =8 51=? F F 97 4 =34 = F 7 + F F 2 F =21 =5 =55 F =34 =8 =89 F =13 =144 =55 F =21 =233 =89 F =34 =377 =144 F 2 =1 In this representation F =610 =233 =55 there is no two consecutive F Fibonacci =987 =377 =89 numbers F =1597 =610 =144 F =233 F 7 =13 17= F 7 +4 F 4 =3 4= F 4 +1 The Fibonacci sequence F 1 =1, F 2 =1, F 3 =2, F 4 =3, F 5 =5, F 6 =8, F 7 =13 F 8 =21, F 9 =34, F 10 =55, F 11 =89, F 12 =144, F 13 =233, F 14 =377, F 15 =610, The sequence of integers 1=F 2, 2=F 3, 3=F 4, 4= F 4 + F 2, 5=F 5, 6= F 5 + F 2, 7= F 5 + F 3, 8= F 6, 9= F 6 + F 2, 10= F 6 + F 3, 11= F 6 + F 4, 12= F 6 +F 4 +F 2

5 The Fibonacci sequence F 1 =1, F 2 =1, F 3 =2, F 4 =3, F 5 =5, F 6 =8, F 7 =13, F 8 =21, F 9 =34, F 10 =55, F 11 =89, F 12 =144, F 13 =233, F 14 =377, F 15 =610,... Divisibility (Lucas, 1878) If b!3, divides then a, b divides then F b a divides if and only F a. if F b divides F a. Examples: F 12 =144 is divisible by F 3 =2, F 4 =3, F 6 =8, F 14 =377 by F 7 =13, F 16 =987 by F 8 =21. Analogy with the sequence 2 n 2 b divides 2 a if and only if b"a. Sequence u n = 2 n b -1 divides 2 a -1 if and only if b divides a. If a=kb set x= 2 b so that 2 a = x k and write x k -1 =(x -1)(x k-1 + x k-2 + +x+1) Recurrence relation : u n+1 = 2 u n + 1 Exponential Diophantine equations Y. Bugeaud, M. Mignotte, S. Siksek (2004): The only perfect powers in the Fibonacci sequence are 1, 8 and 144. Exponential Diophantine equations T.N. Shorey, TIFR (2005): The product of 2 or more consecutive Fibonacci numbers other than F 1 F 2 is never a perfect power. Equation: F n =a b Unknowns: n, a and b with n!1, a!1 and b!2. Conference DION2005, TIFR Mumbai, december 16-20, 2005

6 Phyllotaxy Leaf arrangements Study of the position of leaves on a stem and the reason for them Number of petals of flowers: daisies, sunflowers, aster, chicory, asteraceae, Spiral patern to permit optimum exposure to sunlight Pine-cone, pineapple, Romanesco cawliflower, cactus Université de Nice, Laboratoire Environnement Marin Littoral, Equipe d'accueil "Gestion de la Biodiversité" Phyllotaxy

7 Geometric construction of the Fibonacci sequence 8 This is a nice rectangle A square A N I C E x=1/x 1 x R E C T A N G L E

8 The Golden Number Fra Luca Pacioli (1509) De Divina Proportione 5= Exercise: The Golden Rectangle 1/" 1 1/" 2 " 1/" 3

9 Ammonite (Nautilus shape) Spirals in the Galaxy The Golden Number in art, architecture, aesthetic

Kees van Prooijen http://www.kees.cc/gldsec.

Johnson Automatic Music for six percussionists Phyllotaxy J.

10 Kees van Prooijen Music and the Fibonacci sequence Dufay, XV ème siècle Roland de Lassus Debussy, Bartok, Ravel, Webern Stoskhausen Xenakis Tom Johnson Automatic Music for six percussionists Phyllotaxy J. Kepler (1611) uses the Fibonacci sequence in his study of the dodecahedron and the icosaedron, and then of the symmetry of order 5 of the flowers Stéphane Douady et Yves Couder Les spirales végétales La Recherche 250 (janvier 1993) vol. 24. nbor7.gif Regular pentagons and dodecagons " =2 cos(#/5)

$Diffraction of$ tessalation

11 Penrose non-periodic tiling patterns and quasi-crystals proportion=" Diffraction of quasi-crystals Doubly periodic tessalation (lattices) - cristallography Géométrie d'un champ de lavande François Rouvière (Nice)

12 ExplosionDesMathematiques/ Presentation.html L explosion des Mathématiques

Lecture 8: Phyllotaxis, the golden ratio and the Fibonacci sequence

The 77 th Compton lecture series Frustrating geometry: Geometry and incompatibility shaping the physical world Lecture 8: Phyllotaxis, the golden ratio and the Fibonacci sequence Efi Efrati Simons postdoctoral