  Currently there is no known efficient formula for primes. Besides that, prime
numbers have great importance in e.g., information technology such as
public-key cryptography, and their position and possible or impossible
functional generation among the natural numbers is an ancient dilemma. The
properties of the functions 2ab+a+b in the domain of natural numbers are
introduced, analyzed, and exhibited to illustrate how these single out all the
prime numbers from the full set of odd numbers. The characterization of odd
primes vs. odd non-primes can be done with 2ab+a+b among the odd natural
numbers as an analogue to the other, well known type of fundamental
characterization for irrational and rational numbers among the real numbers.
The prime number theorem, twin primes and erratic nature of primes, are also
commented upon with respect to selection, as well as with the Fermat and Euler
numbers as examples.
