The Fundamental Theorem of Arithmetic: With an Elementary Introduction to Prime Numbers

A prime number is a natural number greater than 1, which cannot be expressed as a product of two smaller natural numbers. Stated differently, a prime number is a number divided only by 1 and by itself. No other natural number can divide a prime number. For example, the numbers: 2, 3, 5, 7, 11, 13, 17, 19, etc, are prime numbers. All the other numbers are called composite numbers. For example, the numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, etc, are composite numbers. The prime number 7 is divided by 1 and 7 only, while the composite number 12 is divided by 1, by 12, but also, by 2, and by 3, and by 4, and by 6.
Prime numbers, despite their simple definition, still hide many secrets, and their intensive study has been the subject of many eminent mathematicians, for more than 25 centuries. Prime numbers and their properties were studied, for the first time, by the ancient Greeks. Euclid defined a prime number as “a number measured by a unit alone” and also gave the first proof that there are infinitely many prime numbers, (around 300 BC). Eratosthenes of Cyrene, around 200 BC, devised a procedure, (an algorithm), to find all the prime numbers less than a given natural number, (the sieve of Eratosthenes).
The main problem with prime numbers is that the way they appear is highly irregular. There is no known formula, or method, to predict the next prime number from the preceding one. Great mathematicians, Gauss, Euler and Riemann among others, have made important contributions to the prime numbers, but, the main problem of finding a pattern as how the prime numbers are distributed, still, remains unsolved. There are also, many other unsolved problems concerning the prime numbers. One of them is the Goldbach’s conjecture, that every even number greater than 2 can be expressed as the sum of two primes. Another famous unsolved problem is the twin prime conjecture, that there are infinitely many pairs of primes that differ by 2, (like 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc). Other unsolved problems concerning prime numbers and some historical remarks, like, Euclid’s theorem, the sieve of Eratosthenes of Cyrene, Fermat numbers, etc, are discussed in section 3-4.
Primes are of fundamental importance in number theory, because of the fundamental theorem of arithmetic: every natural number greater than 1, can be expressed as a product of prime numbers, and this product representation is unique. In this little book we provide a rigorous proof of this theorem and present some of its applications, (like for instance, to find the number of divisors and the sum of the divisors of any natural number, the perfect numbers and the amicable numbers).
The material covered in the book is shown analytically in the table of contents. A comprehensive introduction to the prime numbers and some of its more important properties are presented, which are essential for the proof of the fundamental theorem.
Solving problems with prime numbers is not an easy task. Most of the times, the solution requires ingenuity and imagination. Some other times, when a direct method of approach does not lead to the solution of the problem, we may proceed by contradiction, which is a popular and powerful method when working with prime numbers. The careful reader will find that many of the solved exercises have been solved by contradiction.
The book contains 60 illustrative, solved exercises and 115 problems for solution.