Chapter 11

The Number Theory Revival• Between Diophantus and Fermat

• Fermat’s Little Theorem

• Fermat’s Last Theorem

11.1 Between Diophantus and Fermat

• China, Middle Ages (11th -13th centuries)– Chinese Remainder Theorem– Pascal’s triangle

• Levi ben Gershon (1321)– “combinatorics and mathematical induction”

(formulas for permutations and combinations)

• Blaise Pascal (1654)– unified the algebraic and combinatorial

approaches to “Pascal’s triangle”

Pascal’s trianglein Chinese mathematics

The Chineseused Pascal’s triangle to

find the coefficients of (a+b)n

Pascal’s Triangle

542332455

4322344

3223

222

1

0

510105 )(

464)(

333)(

2)(

)(

1)(

babbababaaba

babbabaaba

babbaaba

bababa

baba

ba

15101051)5(

14641)4(

1331)3(

121)2(

11)1(

1)0(

• As we know now, the kth element of nth row is since

• Thus Pascal’s triangle expresses the following property of binomial coefficients:

!)!(

!

kkn

n

k

n

n

k

kknn

k

kknn bakkn

nba

k

nba

00 !)!(

!)(

k

n

k

n

k

n 1

1

1

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

nn

k

kknnk

nk

n

nn

k

kknnk

n

k

kknnk

n

bbaCCa

bbaCbaCa

1

1

11

1

1

1

11

1

1

1

1

0

1)1(11

0

1

1

0

)1(11

0

)1(1

111 )()()()()(

n

j

jjnnj

n

j

jjnnj

n

j

jjnnj

n

j

jjnnj

nnnn

baCbaC

bbaCabaC

bbaababababa

n

k

kknnk

n baCba0

)(

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

n

k

kknnk

n

j

jjnnj baCbaC

1

11

1

0

1

• Knowing that C0n-1=1, Cn-1

n-1 =1 and replacing j by k in the

first sum we obtain

11

1

nk

nk

nk CCC

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

!)!(

!

kkn

n

k

n

Why “Pascal’s 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

Pierre de Fermat

Born: 1601 in Beaumont (near Toulouse, France)

Died: 1665 in Castres (France)

11.2 Fermat’s 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: Fermat’s 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

• Fermat’s original proof of the theorem is unknown

Proof

• Proof can be conducted in two alternative ways:

– iterated use of binomial theorem

– application of the following multinomial theorem:

m

m

km

kk

nkkk m

nm aaa

kkk

naaa

21

21

2121

21 !!!

!)(

11.3 Fermat’s 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 Bachet’s translation of Diophantus’ “Arithmetica”

• Theorem There are no triples (a,b,c) of positive integers such thatan + bn = cn where n > 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 < 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

• First significant step (after Kummer):Proof of Mordell’s conjecture (1922) about algebraic curves given by Falting (1983)

• Applied to the “Fermat curve” xn + yn = 1 for n > 3, this conjecture provides the following statement– Fermat curve contains at most finitely many of rational

points for each n > 3• Therefore, Falting’s result imply that equation an + bn = cn

can have at most finitely many solutions for each n > 3• The complete proof of Fermat’s 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)