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

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 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   n   P hcf 45, 1 100 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Two positive integers, m and n , are chosen at random.     m n  P hcf , 1 ? 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. 1 The probability that 2 divides both m and n is 2 2 The probability that 2 doesn’t divide at least one of m and n is 1  1 . 2 2 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Two positive integers, m and n , are chosen at random. 1 The probability that 3 divides both m and n is 3 2 The probability that 3 doesn’t divide at least one of m and n is 1  1 . 2 3 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

Two positive integers, m and n , are chosen at random. 1 The probability that a given prime p divides both m and n is p 2 The probability that p doesn’t divide at least one of m and n is 1  1 . 2 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       1 1 1 1 1            1 1 1 1 1 ...       2 2 2 2 2 2 3 5 7 11 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

1 3    1 0.75 2 2 4    1 1 3 8 2          1 1 0.667    2 2 2 3 4 9 3     1 1 1 3 8 24 16             1 1 1 0.64     2 2 2 2 3 5 4 9 25 25      1 1 1 1 3 8 24 48               1 1 1 1 0.627      2 2 2 2 2 3 5 7 4 9 25 49 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

The Basel Problem (1735) 1 1 1 1     ... 2 2 2 2 1 2 3 4 Leonhard Euler 1707 - 1783 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

The Basel Problem 1 1 1 1     ... 2 2 2 2 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 f sin ... f 0 0 x x a a x a x a x a x a 0 1 2 3 4 0         f f 0 x         f f 0 x         f f 0 x MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

             2 3 4 f sin ... f 0 0 x x a a x a x a x a x a 0 1 2 3 4 0               2 3 f cos 2 3 4 ... f 0 1 x x a a x a x a x a 1 2 3 4 1         f f 0 x         f f 0 x MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

             2 3 4 f sin ... f 0 0 x x a a x a x a x a x a 0 1 2 3 4 0               2 3 f cos 2 3 4 ... f 0 1 x x a a x a x a x a 1 2 3 4 1               2 f sin 2 6 12 ... f 0 0 2 x x a a x a x a 2 3 4 2         f f 0 x MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

             2 3 4 f sin ... f 0 0 x x a a x a x a x a x a 0 1 2 3 4 0               2 3 f cos 2 3 4 ... f 0 1 x x a a x a x a x a 1 2 3 4 1               2 f sin 2 6 12 ... f 0 0 2 x x a a x a x a 2 3 4 2               f cos 6 24 ... f 0 1 6 x x a a x a 3 4 3 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

3 5 7 9 x x x x       sin ... x x 3! 5! 7! 9! MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

   2 5 6 0 x x Equivalent       2 3 0 quadratic x x equations?       2 3 0 x x    x x       1 1 0    2 3 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 x x x x       sin ... x x 3! 5! 7! 9! 2 4 6 8 sin x x x x x       1 ... 3! 5! 7! 9! x MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

3 5 7 9 x x x x       sin ... x x 3! 5! 7! 9! 2 4 6 8 sin x x x x x       1 ... 3! 5! 7! 9! x             x x x x x x                    1 1 1 1 1 1 ...                            2 2 3 3       MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

3 5 7 9 x x x x       sin ... x x 3! 5! 7! 9! 2 4 6 8 sin x x x x x       1 ... 3! 5! 7! 9! x             x x x x x x                    1 1 1 1 1 1 ...                            2 2 3 3           2 2 2 x x x         1 1 1 ...    2 2 2     4 9 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

        1 1 1 1 1                   ...             2 2 2 2 6 4 9 16  2 1 1 1 1      ... 2 2 2 2 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       1 1 1 1 1            1 1 1 1 1 ...       2 2 2 2 2 2 3 5 7 11 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy

     1 1 1 1 1 1 1 1                1 ... 1 1 1 ...      2 2 2 2 2 2 2 2 2 3 4 5 6 2 3 5 MEI 2014 Conference. How rare are co-prime pairs? Bernard Murphy