Concrete mathematics : a foundation for computer science

116 NUMBER THEORY
A fraction m/n is in lowest terms if and only if m I n. Since we
reduce fractions to lowest terms by casting out the largest common factor of
numerator and denominator, we suspect that, in general,
mlgcd(m,n) 1 n/gcd(m, n) ; (4.27)
and indeed this is true. It follows from a more general law, gcd(km, kn) =
kgcd(m, n), proved in exercise 14.
The I relation has a simple formulation when we work with the primeexponent
representations of numbers, because of the gcd rule (4.14):
mln min(m,,n,) = 0 for allp. (4.28)
Furthermore, since mP and nP are nonnegative, we can rewrite this as The dot product is
zero, like orthogonal
mln mPnP = 0 forallp. (4.2g) vectors.
And now we can prove an important law by which we can split and combine
two I relations with the same left-hand side:
klm a n d kin k I mn. (4.30)
In view of (4.2g), this law is another way of saying that k,,mp = 0 and
kpnp = 0 if and only if kP (mp + np) = 0, when mp and np are nonnegative.
There’s a beautiful way to construct the set of all nonnegative fractions
m/n with m I n, called the Stem-Brocot tree because it was discovered Interesting how
independently by Moris Stern [279], a German mathematician, and Achille
Brocot [35], a French clockmaker. The idea is to start with the two fractions
mathematicians
will say “discovered”
when abso-
(y , i) and then to repeat the following operation as many times as desired:
Insert m+m’
n+ between two adjacent fractions z and $ .
The new fraction (m+m’)/(n+n’) is called the mediant of m/n and m’/n’.
For example, the first step gives us one new entry between f and A,
and the next gives two more:
0 11 21
7, 23 7, 7, 5 *
The next gives four more,
0 1 1 2 1 3 2 3 1
7, 3, 2, 3, 7, 2, 7, 7, 8;
lute/y anyone e/se
would have said