Concrete mathematics : a foundation for computer science

134 NUMBER THEORY
For example, (~(12) = cp(4)(p(3) = 292 = 4, because n is prime to 12 if “Sisint A et B nuand
only if n mod 4 = (1 or 3) and n mod 3 = (1 or 2). The four values prime
to 12 are (l,l), (1,2), (3,111, (3,2) in the residue number system; they are
meri inter se primi
et numerus partium
ad A primarum
1, 5, 7, 11 in ordinary decimal notation. Euler’s theorem states that n4 3 1 sjt = a, numerus
(mod 12) whenever n I 12.
vero partium ad B
A function f(m) of positive integers is called mult$icative if f (1) = 1 ~~f~u~e$ raz’
and tium ad productum
AB primarum erit
f(mlm2) = f(m)f(m2) whenever ml I mz. (4’5l) = “‘:L. Euler [#J]
We have just proved that q)(m) is multiplicative. We’ve also seen another
instance of a multiplicative function earlier in this chapter: The number of
incongruent solutions to x’ _=
1 (mod m) is multiplicative. Still another
example is f(m) = ma for any power 01.
A multiplicative function is defined completely by its values at prime
powers, because we can decompose any positive integer m into its primepower
factors, which are relatively prime to each other. The general formula
f(m) = nf(pmpl, if m= rI pmP (4.52)
P P
holds if and only if f is multiplicative.
In particular, this formula gives us the value of Euler’s totient function
for general m:
q(m) = n(p”p -pm,-‘) = mn(l -J-).
P\m P\m r
For example, (~(12) = (4-2)(3- 1) = 12(1 - i)(l - 5).
Now let’s look at an application of the cp function to the study of rational
numbers mod 1. We say that the fraction m/n is basic if 0 6 m < n. Therefore
q(n) is the number of reduced basic fractions with denominator n; and
the Farey series 3,, contains all the reduced basic fractions with denominator
n or less, as well as the non-basic fraction f.
The set of all basic fractions with denominator 12, before reduction to
lowest terms, is
Reduction yields