Multiplicative Number Theory and Problems about Primes

Our first topic this semester will be about the properties of the positive integers under the operations of multiplication and division. The main problems arise from the fact that the quotient of two integers is not generally an integer. This leads to the following definition. Given two integers a and b, we say that b divides a, written a|b, if there is another integer c such that a=bc. If b divides a, we call b a divisor of a. All integers are divisors of 0. All integers a are divisible by $\pm 1$ and $\pm a$. If a has any other divisors, we call a composite. Otherwise, we call a prime, unless $a=\pm 1$ which are called units. Of the positive integers not greater than 20, the composite numbers are

$$\disp 4=2\cdot 2,\;6=2\cdot 3,\;8=2\cdot 4,\;9=3\cdot 3,\;10=... ...dot 7,\;15=3\cdot 5,\;16=4\cdot 4,\;18=2\cdot 9,\;20=4\cdot 5.$$

The prime numbers less than 20 are

$$2,\;3,\;5,\;7,\;11,\;13,\;17,\;19.$$