Code and ciphers: Julius Caesar, the Enigma and the internet

192
appendix
Suppose that we have already looked at n people and that no two of
them have the same birthday. Then when we look at the (n�1)st person
the probability that he/she will not have a birthday in common with any
of the others is
(365 � n)
365
The probability that no pair of people among 23 people chosen at random
will have the same birthday is therefore
364
365 �363
365 �362
365 �•••�343
365
the value of which (to three decimal places ) is 0.493. Therefore the probability
that at least one pair will have the same birthday is (1�0.493),
which is 0.507, and since this number is greater than a half there is a
better than evens chance of there being a pair with the same birthday. Had
we confined our attention to 22 people, rather than 23, the probability of
there being at least one pair with the same birthday would have been less
than a half, 0.476 to three decimal places.
Chapter 3
M4 Euclid’s proof that there are an infinite number of
primes
Suppose, on the contrary, that there are only a finite number of primes
and that they are
2, 3, 5, 7, 11, ..., P,
Consider the number, N, formed by multiplying all of them together and
adding 1:
N�2�3�5�7�11�•••�P�1.
Clearly N is not divisible by 2 or 3 or 5 or 7 or 11 or ... or P since it leaves
remainder 1 on each such division. So N is not divisible by any prime in
the list. It is therefore either itself a prime or divisible by some prime
which is not in the list, and in either case our alleged list of all the
primes must be incomplete. There are therefore an infinite number of
primes.