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

Chapter 13
M21 (Rate of increase of the number of primes)
The Prime Number Theorem [12.1] tells us that as N increases the number
of primes less than N, which is traditionally denoted by �(N), is asymptotically
approximated by
N
�(N) �
log(N)
The logarithm being to base e.
It follows that as N increases the fraction of integers less than N which are
primes slowly decreases. By studying tables of the number of primes less than
1000, 10 000, 100 000 Gauss discovered the Prime Number Theorem in
1793, but was unable to prove it. The relevant data are shown in Table A.2.
Table A.2
N Number of primes less than N Fraction of numbers which are prime
1000 168 1 in 15.95
10 000 1 229 1 in 18.14
100 000 9 592 1 in 10.43
1000 000 78 498 1 in 12.74
If we now difference the numbers in the right-hand column we get
18.14�15.95�2.19,
10.43�18.14�2.29,
12.74�10.43�2.31,
and Gauss conjectured that this difference would be essentially constant
as N increased and would be approximately 2.3. Now log(10) is approximately
equal to 2.3 and this implied that if we increase N by a factor of 10
the reciprocal of the fraction of integers less than N which are primes
increases by log(10), a statement which is equivalent to the Prime Number
Theorem. Gauss’s conjecture was correct but it was more than a hundred
years before the Prime Number Theorem was proved. See also [12.1].
M22 Calculating remainder using modular arithmetic
(1) That (59) 96 is a number with 171 digits follows from the fact that
96log 10 (59)�96�(1.770 85...)�170.0018...
so (59) 96 lies between 10 170 and 10 171 and therefore has 171 digits.
Mathematical aspects 209