Discovering primes
Unhappily there are no set formulae for identifying primes, and there seems to
be no pattern in their appearances among the whole numbers. One of the first
methods for finding them was developed by a younger contemporary of
Archimedes who spent much of his life in Athens, Erastosthenes of Cyrene. His
precise calculation of the length of the equator was much admired in his own
time. Today he’s noted for his sieve for finding prime numbers. Erastosthenes
imagined the counting numbers stretched out before him. He underlined 2 and
struck out all multiples of 2. He then moved to 3, underlined it and struck out all
multiples of 3. Continuing in this way, he sieved out all the composites. The
underlined numbers left behind in the sieve were the primes.
So we can predict primes, but how do we decide whether a given number is a
prime or not? How about 19,071 or 19,073? Except for the primes 2 and 5, a
prime number must end in a 1, 3, 7 or 9 but this requirement is not enough to
make that number a prime. It is difficult to know whether a large number ending
in 1, 3, 7 or 9 is a prime or not without trying possible factors. By the way,
19,071 = 3^2 × 13 × 163 is not a prime, but 19,073 is.
Another challenge has been to discover any patterns in the distribution of the
primes. Let’s see how many primes there are in each segment of 100 between 1
and 1000.