8/3/2019 102 Week 1

1/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

Hilberts Hotel

8/3/2019 102 Week 1

2/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

8/3/2019 102 Week 1

3/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

n n+1

8/3/2019 102 Week 1

4/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

8/3/2019 102 Week 1

5/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

n 2n

8/3/2019 102 Week 1

6/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

n 2n

8/3/2019 102 Week 1

7/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

8/3/2019 102 Week 1

8/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .

n 2n

8/3/2019 102 Week 1

9/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

8/3/2019 102 Week 1

10/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

8/3/2019 102 Week 1

11/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

Suppose the opposite. i.e. that there is a natural number which is a

prime and is bigger than any other prime. Call it p.

8/3/2019 102 Week 1

12/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

Suppose the opposite. i.e. that there is a natural number which is a

prime and is bigger than any other prime. Call it p.

Let p1, p2, ..., pn be all the primes smaller than p.

8/3/2019 102 Week 1

13/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

Suppose the opposite. i.e. that there is a natural number which is a

prime and is bigger than any other prime. Call it p.

Let p1, p2, ..., pn be all the primes smaller than p.

Consider the number q = p1 p2 ... pn + 1

8/3/2019 102 Week 1

14/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

Suppose the opposite. i.e. that there is a natural number which is a

prime and is bigger than any other prime. Call it p.

Let p1, p2, ..., pn be all the primes smaller than p.

Consider the number q = p1 p2 ... pn + 1

q is not divisible by any natural number except 1 and q itself.

Therefore, q is a prime.

8/3/2019 102 Week 1

15/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

Suppose the opposite. i.e. that there is a natural number which is a

prime and is bigger than any other prime. Call it p.

Let p1, p2, ..., pn be all the primes smaller than p.

Consider the number q = p1 p2 ... pn + 1

q is not divisible by any natural number except 1 and q itself.

Therefore, q is a prime.

q is also bigger than p.

8/3/2019 102 Week 1

16/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

Suppose the opposite. i.e. that there is a natural number which is a

prime and is bigger than any other prime. Call it p.

Let p1, p2, ..., pn be all the primes smaller than p.

Consider the number q = p1 p2 ... pn + 1

q is not divisible by any natural number except 1 and q itself.

Therefore, q is a prime.

q is also bigger than p.

Therefore, q is a prime bigger than p, which contradicts our

assumption that p is the biggest prime.

8/3/2019 102 Week 1

17/43

There is no biggest prime.

(i.e. the set of all primes is infinite)

Proof byreduction ad absurdum:

Suppose the opposite. i.e. that there is a natural number which is a

prime and is bigger than any other prime. Call it p.

Let p1, p2, ..., pn be all the primes smaller than p.

Consider the number q = p1 p2 ... pn + 1

q is not divisible by any natural number except 1 and q itself.

Therefore, q is a prime.

q is also bigger than p.

Therefore, q is a prime bigger than p, which contradicts our

assumption that p is the biggest prime.

Conclusion: There is no biggest prime. QED

8/3/2019 102 Week 1

18/43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ..

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ..

1st bus: passenger n goes to the room 3n

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 .

8/3/2019 102 Week 1

19/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1st bus: passenger n goes to the room 3n

b1

p1

b1

p2

b1p3

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

8/3/2019 102 Week 1

20/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2nd bus: passenger n goes to the room 5n

b1

p1

b1

p2

b1

p3

b2

p1

b2

p2

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

8/3/2019 102 Week 1

21/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3rd bus: passenger n goes to the room 7n

b1

p1

b1

p2

b1

p3

b2

p1

b2

p2

b3

p1

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

8/3/2019 102 Week 1

22/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 ..

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ..

Bus 4 11 Bus 7 19 Bus 10 31

Bus 5 13 Bus 8 23 Bus 11 37

Bus 6 17 Bus 9 29 Bus 12 41

b1

p1

b1

p2

b1p3

b2

p1

b2

p2

b3

p1

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ..

8/3/2019 102 Week 1

23/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

15 35

21 39

33 45

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 .

8/3/2019 102 Week 1

24/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

15 = 3 5 35 = 5 7

21 = 3 7 39 = 3 11

33 = 3 11 45 = 9 5 = 32 5

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 .

8/3/2019 102 Week 1

25/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3n 5m 3n 13m

3n 7m 3n 17m

3n 11m

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 .

8/3/2019 102 Week 1

26/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3n 5m 3n 13m

3n 7m 3n 17m

3n 11m

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

5n 7m

5n 11m

5n

13m

8/3/2019 102 Week 1

27/43

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

3n 5m 3n 13m

3n 7m 3n 17m

3n 11m

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

5n 7m

5n 11m

5n 13m

3n 5m 7p

3n 5m 7p 19q 43t

8/3/2019 102 Week 1

28/43

8/3/2019 102 Week 1

29/43

8/3/2019 102 Week 1

30/43

8/3/2019 102 Week 1

31/43

8/3/2019 102 Week 1

32/43

8/3/2019 102 Week 1

33/43

8/3/2019 102 Week 1

34/43

8/3/2019 102 Week 1

35/43

8/3/2019 102 Week 1

36/43

8/3/2019 102 Week 1

37/43

8/3/2019 102 Week 1

38/43

8/3/2019 102 Week 1

39/43

8/3/2019 102 Week 1

40/43

8/3/2019 102 Week 1

41/43

8/3/2019 102 Week 1

42/43

5, if the nth digit of rn 5

the nth digit of r0 = 6, if the nth digit of rn 5

8/3/2019 102 Week 1

43/43

5, if the nth digit of rn 5

the nth digit of r0 = 6, if the nth digit of rn 5r0 0.