MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

MEI Conference 2014

How rare are co-prime pairs?

Bernard Murphy bernard.murphy@mei.org.uk

mailto:bernard.murphy@mei.org.uk

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Choose two positive integers at random. The probability that their highest common factor is 1 involves pi squared! This session will explain why. Suitable for all teachers who know about the Maclaurin expansion of sinx.

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Think of a positive integer

less than 100

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Find the highest common factor of your number and 45

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

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

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

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

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

53

P hcf 45, 1 100

n

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Two positive integers, m and n, are chosen at random.

P hcf , 1 ?m n

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

1 2 3 4 5 6 7 8 9 10

1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2 3 1 1 3 1 1 3 1 1 3 1 4 1 2 1 4 1 2 1 4 1 2 5 1 1 1 1 5 1 1 1 1 5 6 1 2 3 2 1 6 1 2 3 2 7 1 1 1 1 1 1 7 1 1 1 8 1 2 1 4 1 2 1 8 1 2 9 1 1 3 1 1 3 1 1 9 1

10 1 2 1 2 5 2 1 2 1 10

63

1 , 10 P hcf , 1 =100

m n m n

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Two positive integers, m and n, are chosen at random.

The probability that 2 divides both m and n is 21

2

The probability that 2 doesn’t divide at least one of m and n is

2

11

2 .

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Two positive integers, m and n, are chosen at random.

The probability that 3 divides both m and n is 21

3

The probability that 3 doesn’t divide at least one of m and n is

2

11

3 .

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Two positive integers, m and n, are chosen at random.

The probability that a given prime p divides both m and n is 21

p

The probability that p doesn’t divide at least one of m and n is

2

11

p

.

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

The probability that two positive integers, m and n, selected at random, are relatively prime, is

2 2 2 2 2

1 1 1 1 11 1 1 1 1 ...

2 3 5 7 11

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2

2 2

2 2 2

2 2 2 2

1 31 0.75

2 4

1 1 3 8 21 1 0.667

2 3 4 9 3

1 1 1 3 8 24 161 1 1 0.64

2 3 5 4 9 25 25

1 1 1 1 3 8 24 481 1 1 1 0.627

2 3 5 7 4 9 25 49

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

The Basel Problem (1735)

2 2 2 2

1 1 1 1...

1 2 3 4

Leonhard Euler 1707 - 1783

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

The Basel Problem

2 2 2 2

1 1 1 1...

1 2 3 4

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 3 4

0 1 2 3 4 0f sin ... f 0 0

f f 0

f f 0

f f 0

x x a a x a x a x a x a

x

x

x

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 3 4

0 1 2 3 4 0

2 3

1 2 3 4 1

f sin ... f 0 0

f cos 2 3 4 ... f 0 1

f f 0

f f 0

x x a a x a x a x a x a

x x a a x a x a x a

x

x

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 3 4

0 1 2 3 4 0

2 3

1 2 3 4 1

2

2 3 4 2

f sin ... f 0 0

f cos 2 3 4 ... f 0 1

f sin 2 6 12 ... f 0 0 2

f f 0

x x a a x a x a x a x a

x x a a x a x a x a

x x a a x a x a

x

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 3 4

0 1 2 3 4 0

2 3

1 2 3 4 1

2

2 3 4 2

3 4 3

f sin ... f 0 0

f cos 2 3 4 ... f 0 1

f sin 2 6 12 ... f 0 0 2

f cos 6 24 ... f 0 1 6

x x a a x a x a x a x a

x x a a x a x a x a

x x a a x a x a

x x a a x a

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

3 5 7 9

sin ...3! 5! 7! 9!

x x x xx x

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 5 6 0

2 3 0

2 3 0

1 1 02 3

x x

x x

x x

x x

Equivalent quadratic

equations?

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

3 5 7 9

2 4 6 8

sin ...3! 5! 7! 9!

sin1 ...

3! 5! 7! 9!

x x x xx x

x x x x x

x

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

3 5 7 9

2 4 6 8

sin ...3! 5! 7! 9!

sin1 ...

3! 5! 7! 9!

1 1 1 1 1 1 ...2 2 3 3

x x x xx x

x x x x x

x

x x x x x x

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

3 5 7 9

2 4 6 8

2 2 2

2 2 2

sin ...3! 5! 7! 9!

sin1 ...

3! 5! 7! 9!

1 1 1 1 1 1 ...2 2 3 3

1 1 1 ...4 9

x x x xx x

x x x x x

x

x x x x x x

x x x

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 2 2 2

2

2 2 2 2

1 1 1 1 1...

6 4 9 16

1 1 1 1...

6 1 2 3 4

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

The probability that two positive integers, m and n, selected at random, are relatively prime, is

2 2 2 2 2

1 1 1 1 11 1 1 1 1 ...

2 3 5 7 11

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 11 ... 1 1 1 ...

2 3 4 5 6 2 3 5

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

The probability that two randomly chosen positive integers, m and n, are relatively prime, is

2 2 2 2 2

2 2

2 2 2 2 2

1 1 1 1 11 1 1 1 1 ...

2 3 5 7 11

1 1 6

1 1 1 1 11 ...

2 3 4 5 6 6

0.60793

MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

2 4 6 8 2 2 2 2

2 2 2 21 ... 1 1 1 1 ...

3! 5! 7! 9! 4 9 16

x x x x x x x x

Coefficient of 2 :x

2

2 2 2 2

1 1 1 1...

6 1 2 3 4

Coefficient of 4 ?x