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Apart from anything else, the Greeks proved that the number of prime numbers is infinite. The proof is simple.
     
First, note that every number can be expressed as the product of prime factors of the number.
Multiply n prime numbers and add 1 to the product. We get a number that we denote by M .
If M is a prime number, we have found one not among the n prime numbers that we had started with.
If M is not prime, consider a prime factor of M .
The prime factor can be divided into M without a remainder, hence the prime factor of M is different from each of the n prime numbers we started with as these, when divided into M , give a remainder 1.
Thus, in the second possibility (i.e., step d), we also found another prime number in addition to the n numbers that we multiplied.
We have thus shown that there are more than n prime numbers. But n is an arbitrary number, therefore the number of prime numbers is not finite and the proof is complete.
    But why should anyone be interested in the question of whether the number of prime numbers is infinite? Where in evolution would the question whether there is an infinite number of any particular object be a meaningful question? Interest in the mathematical properties of prime numbers, including apparently useless properties, started with the Greeks, continued throughout generations of mathematicians, and is still today an important part of mathematical research. In current times uses of prime numbers have been discovered apart from the abstract mathematical interest, including commercial uses such as encoding, which we will discuss further on. For thousands of years the interest was purely mathematical. For the Greeks, however, it seems that the involvement in numbers was not simply motivated by curiosity but by the belief that thus they would better understand the world around them.
    The next leap in the amazing change wrought by the Greeks in the development of mathematics is attributed to the Academy of Athens and its disciples, and in particular to its founder, Plato, his friend Eudoxus, and Plato's pupil Aristotle. The conceptual contribution of this group may be summarized as the formulation of the approach that bases mathematics on axioms and on logic as the essential tool in the system of deductive proof. As we will try to establish here and later in this book, these two contributions conflict with the natural intuition of human thought.

    Plato (427–347 BCE) came from an aristocratic and influential family. He was a pupil of Socrates, who is considered the father of general and political Western philosophy. In his youth, Plato entertained political ambitions, but he abandoned them, perhaps because he saw what happened to Socrates, who was sentenced to death for his opposition to and criticism of the rulers of Athens. Plato traveled widely in the ancient world, visiting Egypt and the Greek colonies in Sicily, where he became acquainted with Egyptian mathematics and the Pythagoreans. On returning to Athens, he founded the first academy in the Western world. The academy had a decisive influence on contemporary science and philosophy. Plato was essentially a philosopher, and his interest in mathematics stemmed from his belief that the truth about the nature of science can be revealed only via mathematics. On the entrance of the academy he inscribed “Let none but geometers enter here.” Plato went further and, in accordance with the philosophy he developed in other fields, claimed that mathematics, or mathematical results, have an independent existence in the world of ideas that are not necessarily related to the earthly reality that we experience in daily life. Specifically, we do not invent mathematical results; we discover them. The right way to do this is to formulate the assumptions, which we will call axioms, and to use them to draw mathematical truths from them using deductive logic. To this end the axioms should be simple and self-explanatory. The smaller the number of axioms,