Number Theory is the study of the properties and relations of the integers. There are many sets of positive integers of particular interest, such as the primes and the perfect numbers. An important area in number theory is the analysis of prime numbers. A prime number is an integer p>1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, …. The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product (a = p1 • p2 • p3 • • • pn) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 • 2 •5, and it is unique (disregarding order) since 20 has this and only this product of primes. Modern number theory begins with Pierre de Fermat, inspired in part by his study of Diophantus.
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