Chapter 11 The Number Theory Revival

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Chapter 11The Number Theory Revival

• Between Diophantus and Fermat
• Fermats Little Theorem
• Fermats Last Theorem

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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
  (formulas for permutations and combinations)
• Blaise Pascal (1654)
• unified the algebraic and combinatorial
  approaches to Pascals triangle

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Pascals trianglein Chinese mathematics
The Chineseused Pascals triangle to find the
coefficients of (ab)n
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Pascals Triangle
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• As we know now, the kth element of nth row
  is since
• Thus Pascals triangle expresses the following
  property of binomial coefficients

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• Indeed, suppose that for all n we have
• Then

• Letting k j 1
• in the second
• sum we get

• Knowing that C0n-11, Cn-1n-1 1 and replacing
  j by k in the
• first sum we obtain

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Combinations, permutations,and mathematical
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 tomathematical induction

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Why Pascals Triangle ?

• Pascal 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
  probabilities

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Pierre de Fermat Born 1601 in Beaumont (near
Toulouse, France) Died 1665 in Castres (France)
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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 ornp n is divisible by p (always)
• Note Fermats Little Theorem turned out to be
  very important for practical applications it is
  an important part in the design of RSA code!
• Fermat was interested in the expressions of the
  form2m 1 (in connection with perfect numbers)
  and, at the same time, he was investigating
  binomial coefficients
• Fermats original proof of the theorem is unknown

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Proof

• Proof can be conducted in two alternative ways
• iterated use of binomial theorem
• application of the following multinomial theorem

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11.3 Fermats Last Theorem
On the other hand, it is impossible for a cube
to be written as a sum of two cubes or a fourth
power to be written as a sum of two fourth powers
or, in general, for any number which is a power
higher than second to be written as a sum of two
like powers. I have a truly marvellous
demonstration of this proposition which this
margin is too small to contain. written by
Fermat in the margin of his copy of Bachets
translation of Diophantus Arithmetica
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• Theorem There are no triples (a,b,c) of positive
  integers such thatan bn cn where n gt 2 is an
  integer
• Proofs for special cases
• Fermat for n 4
• Euler for n 3
• Legendre and Dirichlet for n 5,
• Lame for n 7
• Kummer for all prime n lt 100 except 37, 59, 67
• Note it is sufficient to prove theorem for all
  prime exponents (except 2) and for n 4, since
  if n mp where p is prime and an bn cn
  then(am)p (bm)p (cm)p

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• First significant step (after Kummer)Proof of
  Mordells conjecture (1922) about algebraic
  curves given by Falting (1983)
• Applied to the Fermat curve xn yn 1 for n gt
  3, this conjecture provides the following
  statement
• Fermat curve contains at most finitely many of
  rational points for each n gt 3
• Therefore, Faltings result imply that equation
  an bn cn can have at most finitely many
  solutions for each n gt 3
• The complete proof of Fermats Last Theorem is
  due to Andrew Wiles and follows from much more
  general statement (first announcement in 1993,
  gap found, filled in 1994, complete proof
  published in 1995)