MEI Conference 2014 How rare are co-prime pairs? Bernard Murphy - - PowerPoint PPT Presentation

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mei conference 2014

MEI Conference 2014 How rare are co-prime pairs? Bernard Murphy - - PowerPoint PPT Presentation

Mar 19, 2023 457 likes •781 views

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

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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

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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

f sinx.

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

Think of a positive integer less than 100

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

Find the highest common factor

f your number and 45

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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  

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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 

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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   

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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

2

1 2

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

2

1 1 2 

.

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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

2

1 3

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

2

1 1 3  .

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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

2

1 p

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

2

1 1 p 

.

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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 1 1 1 1 1 1 ... 2 3 5 7 11                       

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

2 2 2 2 2 2 2 2 2 2

1 3 1 0.75 2 4 1 1 3 8 2 1 1 0.667 2 3 4 9 3 1 1 1 3 8 24 16 1 1 1 0.64 2 3 5 4 9 25 25 1 1 1 1 3 8 24 48 1 1 1 1 0.627 2 3 5 7 4 9 25 49                                                              

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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

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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    

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

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

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

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

               

2 3 4 1 2 3 4

f sin ... f 0 f f f f f f x x a a x a x a x a x a x x x                     

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

               

2 3 4 1 2 3 4 2 3 1 2 3 4 1

f sin ... f 0 f cos 2 3 4 ... f 1 f f f f x x a a x a x a x a x a x x a a x a x a x a x x                           

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

               

2 3 4 1 2 3 4 2 3 1 2 3 4 1 2 2 3 4 2

f sin ... f 0 f cos 2 3 4 ... f 1 f sin 2 6 12 ... f 2 f f 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                                 

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

               

2 3 4 1 2 3 4 2 3 1 2 3 4 1 2 2 3 4 2 3 4 3

f sin ... f 0 f cos 2 3 4 ... f 1 f sin 2 6 12 ... f 2 f cos 6 24 ... f 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                                       

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

3 5 7 9

sin ... 3! 5! 7! 9! x x x x x x      

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

     

2

5 6 2 3 2 3 1 1 2 3 x x x x x x x x                     

Equivalent quadratic equations?

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

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

3 5 7 9 2 4 6 8

sin ... 3! 5! 7! 9! sin 1 ... 3! 5! 7! 9! x x x x x x x x x x x x            

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

     

3 5 7 9 2 4 6 8

sin ... 3! 5! 7! 9! sin 1 ... 3! 5! 7! 9! 1 1 1 1 1 1 ... 2 2 3 3 x x x x x x x x x x x x x x x x x x                                                                      

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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! sin 1 ... 3! 5! 7! 9! 1 1 1 1 1 1 ... 2 2 3 3 1 1 1 ... 4 9 x x x x x x x x x x x x x x x x x x x x x                                                                                         

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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                                            

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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 1 1 1 1 1 1 ... 2 3 5 7 11                       

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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 1 1 ... 1 1 1 ... 2 3 4 5 6 2 3 5                         

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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 1 1 1 1 1 1 ... 2 3 5 7 11 1 1 6 1 1 1 1 1 1 ... 2 3 4 5 6 6 0.60793                                         

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

2 4 6 8 2 2 2 2 2 2 2 2

1 ... 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