The search for prime numbers is one of the most paradoxical problems in mathematics.  Scientists have been trying to solve it for several millennia, but, acquiring new versions and hypotheses, this mystery still remains unsolved.  The appearance of prime numbers is not subject to any system: they arise spontaneously in a series of natural numbers, ignoring all attempts by mathematicians to identify patterns in their sequence.  This book will allow the reader to trace the evolution of scientific concepts from ancient times to the present day and will introduce the most interesting theories of searching for prime numbers.