I would state the obvious if I said that primes are special numbers. But what makes primality so special? Don Zagier, an eminent specialist, said: Upon looking at these numbers, one has the feeling of being in the presence of one of the inexplicable secrets of creation. That's awesome, isn't it? In antiquity, the Greeks already tried to understand the patterns underpinning the prime numbers. A way to visualize those patterns is through prime number sieves. Eratosthenes is said to have proposed the first sieve, the Eratosthenes sieve. (Hey, have you noticed that Eratosthenes was born 2287 years ago? So we are in year 2287 of the Eratosthenian era. Also a prime number). But there are other sieves. More of that later.
Prime numbers occur in nature in cyclic processes. Imagine that, somewhere in the universe, there's a star with a huge number of planets, all of them having the same orbiting speed. They are numbered from 1 to N, planet n°1 being the inner planet and the diameters of their circular orbits are as following:
- The diameter of planet n°2's orbit is twice the diameter of planet n°1's orbit.
- The diameter of planet n°3's orbit is three times the diameter of planet n°1's orbit.
- The diameter of planet n°4's orbit is four times the diameter of planet n°1's orbit.
- And so on, the unit length being the diameter of planet n°1's orbit, the nth planet has an orbit of diameter n, n being an integer between 1 and N.
The succession of circles n°5 together with n°2 and 3 sieves out all primes between 5 and 7², see Figure 5.
Euler's P(n) = n² − n + 41):
n²+21n+1This way of sieving has been described earlier and more extensively by physicist Imre Mikoss. Check his work at "The Prime Numbers Hidden Symmetric Structure and its Relation to the Twin Prime Infinitude and an Improved Prime Number Theorem".
Happy prime year!