Chapter 11The Number Theory Revival.ppt
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1、Chapter 11 The Number Theory Revival,Between Diophantus and Fermat Fermats Little Theorem Fermats Last Theorem,11.1 Between Diophantus and Fermat,China, Middle Ages (11th -13th centuries) Chinese Remainder Theorem Pascals triangle Levi ben Gershon (1321) “combinatorics and mathematical induction” (f
2、ormulas for permutations and combinations) Blaise Pascal (1654) unified the algebraic and combinatorial approaches to “Pascals triangle”,Pascals triangle in Chinese mathematics,The Chinese used Pascals triangle to find the coefficients of (a+b)n,Pascals Triangle,As we know now, the kth element of nt
3、h row is since Thus Pascals triangle expresses the following property of binomial coefficients:,Indeed, suppose that for all n we have Then,Letting k = j + 1in the secondsum we get,Knowing that C0n-1=1, Cn-1n-1 =1 and replacing j by k in thefirst sum we obtain,Combinations, permutations, and mathema
4、tical induction,Levi ben Gershon (1321) gave the formula for the number of combinations of n things taken k at a time: He also pointed out that the number of permutations of n elements is n! The method he used to show these formulas is very close to mathematical induction,Why “Pascals Triangle” ?,Pa
5、scal demonstrated (1654) that the elements of this triangle can be interpreted in two ways: algebraically as binomial coefficients combinatorially as the number of combinations of n things taken k at a time As application he solved problem of division of stakes and founded the mathematical theory of
6、 probabilities,Pierre de FermatBorn: 1601 in Beaumont (near Toulouse, France) Died: 1665 in Castres (France),11.2 Fermats Little Theorem,Theorem (Fermat, 1640) If p is prime and n is relatively prime to p (i.e. gcd (n,p)=1) then np 1 1 mod p Equivalently, np-1 1 is divisible by p if gcd (n,p)=1 or n
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