Concrete mathematics : a foundation for computer science

4.1 DIVISIBILITY 105
(4.8) works also when n is negative. (In such cases, the nonzero terms on the
right occur when k is the negative of a divisor of n.)
Moreover, a double sum over divisors can be “interchanged” by the law
t x ak,m = x x ak,kl .
m\n k\m k\n L\in/kl
For example, this law takes the following form when n = 12:
al,1 + (al.2 + a2,2) + (al,3 + a3,3)
+ fall4 + a2,4 + a4,4) + (al.6 + a2,6 + a3,6 + a6,6)
+ tal,12 + a2,l2 + a&12 + a4,12 + a6,12 + a12,12)
= tal.l + al.2 + al.3 + al.4 + al,6 + al.12)
+ ta2,2 + a2.4 + a2,6 + a&12) + (a3,3 + as,6 + CQ12)
+ tad,4 $- q12) + (a6,6 + a6,12) + a12,12.
(4.9)
We can prove (4.9) with Iversonian manipulation. The left-hand side is
x x ak.,[n=iml[m=kll = 7 y ak,kt[n=Ml;
i,l k,m>O j k,1>0
the right-hand side is
x t ok.k~[n=jkl[n/k=mll = t t ak,kt[n=mlkl,
j,m k,l>O m k.1>0
which is the same except for renaming the indices. This example indicates
that the techniques we’ve learned in Chapter 2 will come in handy as we study
number theory.
4.2 PRIMES
A positive integer p is called prime if it has just two divisors, namely
1 and p. Throughout the rest of this chapter, the letter p will always stand
How about the p in for a prime number, even when we don’t say so explicitly. By convention,
‘explicitly’? 1 isn’t prime, so the sequence of primes starts out like this:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, .,
Some numbers look prime but aren’t, like 91 (= 7.13) and 161 (= 7.23). These
numbers and others that have three or more divisors are called composite.
Every integer greater than 1 is either prime or composite, but not both.
Primes are of great importance, because they’re the fundamental building
blocks of all the positive integers. Any positive integer n can be written as a