Concrete mathematics : a foundation for computer science

4.2 PRIMES 107
It’s the factor- Sometimes it’s more useful to state the Fundamental Theorem in another
ization, not the
theorem, that’s
way: Every positive integer can be written uniquely in the form
unique.
n = nP”Y where each np 3 0. (4.11)
P
The right-hand side is a product over infinitely many primes; but for any
particular n all but a few exponents are zero, so the corresponding factors
are 1. Therefore it’s really a finite product, just as many “infinite” sums are
really finite because their terms are mostly zero.
Formula (4.11) represents n uniquely, so we can think of the sequence
(nz, n3, n5, . ) as a number system for positive integers. For example, the
prime-exponent representation of 12 is (2,1,0,0,. . . ) and the prime-exponent
representation of 18 is (1,2,0,0, . ). To multiply two numbers, we simply
add their representations. In other words,
k = mn k, = m,+n, forallp. (4.12)
This implies that
m\n
and it follows immediately that
mp < np for all p, (4.13)
k = gcd(m,n) # k, = min(m,,n,) for allp; (4.14)
k = lcm(m,n) W k, = max(m,,n,) for all p. (4.15)
For example, since 12 = 22 .3’ and 18 = 2’ . 32, we can get their gcd and lcm
by taking the min and max of common exponents:
gcd(12,18) = 2min(2,li .3min(l,21 = 21 .31 = 6;
lcm(12,18) = 2maX(2,1) . 3max(l,2) = 22 .32 = 36.
If the prime p divides a product mn then it divides either m or n, perhaps
both, because of the unique factorization theorem. But composite numbers
do not have this property. For example, the nonprime 4 divides 60 = 6.10,
but it divides neither 6 nor 10. The reason is simple: In the factorization
60 = 6.10 = (2.3)(2.5), the two prime factors of 4 = 2.2 have been split
into two parts, hence 4 divides neither part. But a prime is unsplittable, so
it must divide one of the original factors.
4.3 PRIME EXAMPLES
How many primes are there? A lot. In fact, infinitely many. Euclid
proved this long ago in his Theorem 9: 20, as follows. Suppose there were