Concrete mathematics : a foundation for computer science

4
Number Theory
INTEGERS ARE CENTRAL to the discrete mathematics we are emphasizing
in this book. Therefore we want to explore the theory of numbers, an
important branch of mathematics concerned with the properties of integers.
We tested the number theory waters in the previous chapter, by introducing
binary operations called ‘mod’ and ‘gcd’. Now let’s plunge in and
really immerse ourselves in the subject.
4.1 DIVISIBILITY
We say that m divides n (or n is divisible by m) if m > 0 and the
ratio n/m is an integer. This property underlies all of number theory, so it’s
convenient to have a special notation for it. We therefore write
m\n ++ m > 0 and n = mk for some integer k. (4.1)
(The notation ‘mln’ is actually much more common than ‘m\n’ in current
mathematics literature. But vertical lines are overused-for absolute values,
set delimiters, conditional probabilities, etc. -and backward slashes are
underused. Moreover, ‘m\n’ gives an impression that m is the denominator of
an implied ratio. So we shall boldly let our divisibility symbol lean leftward.)
If m does not divide n we write ‘m!qn’.
There’s a similar relation, “n is a multiple of m,” which means almost
the same thing except that m doesn’t have to be positive. In this case we
simply mean that n = mk for some integer k. Thus, for example, there’s only
one multiple of 0 (namely 0), but nothing is divisible by 0. Every integer is
a multiple of -1, but no integer is divisible by -1 (strictly speaking). These
definitions apply when m and n are any real numbers; for example, 271 is
divisible by 7~. But we’ll almost always be using them when m and n are
integers. After all, this is number theory.
102
In other words, be
prepared to drown.
‘I no integer is
dksible by -1
(strictly speaking).”
-Graham, Knuth,
and Patashnik [131]