PRIME NUMBERS
PART III
SHORT HISTORY OF PRIME NUMBERS (from various sources)
“There are hints in the surviving records of the ancient Egyptians that they had some knowledge
of prime numbers (for instance the Egyptian fraction expansions in the Rhind papyrus). However, the
earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's
Elements (about 300 BC) contain important theorems about primes, including the infinitude of primes
and the fundamental theorem of arithmetic, which explains that any integer greater than one can be
written in a unique product of prime numbers. The Sieve of Eratosthenes, attributed to Eratosthenes, is a
simple method to compute primes.
After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640
Pierre de Fermat stated (without proof) which is called Fermat's little theorem: he conjectured that all
numbers of the form 2 to the power 2n+1 are prime. But he was wrong, because, with n equals 5, this
number is divisible par 641. The French monk Marin Mersenne looked at primes of the form 2p − 1, with
p as a prime. The great mathematician Euler's work in number theory included many results about
primes. He showed the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … is divergent, that is to say doesn’t
tend to a finite number.
At the start of the 19th century, Legendre and Gauss independently conjectured that the number of
primes is slowly increasing. Riemann, in his 1859 paper, was about to find the proof, but these were
Hadamard and de la Vallée Poussin who independently proved the prime number theorem in 1896”
11) Were the Greeks the first to work on primes?
12) Explain the fundamental theorem of arithmetic. Try with 21437 and 26877.
13) In your opinion, what does “divergent” mean? Try to find if the four following sums are divergent :
THE PRIME NUMBER THEOREM
The theorem explains that, as x tends to infinity, the number of primes up to x is asymptotic to
x/ln(x), where ln(x) is the natural logarithm of x. Let π(x) be the prime-counting function that gives the
number of primes less than or equal to x, for any real number x. For example, π(10) = 4 because there
are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The formula, known as the asymptotic
π ( x)
law of distribution of prime numbers, is : lim
=1
x →∞ x / ln( x )
14) Using the formula, find an approximation of the number of primes less than 100, then less than 1000.
15) Compute the following limits (on the blackboard)
Next Week : Test on this Chapter !
π 4→∞→π ² 6 → 2