Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 301 — #309
9.3. Prime Mysteries 301
9.2.4 Properties of the Greatest Common Divisor
It can help to have some basic gcd facts on hand:
Lemma 9.2.6.
a) gcd.ka; kb/ D k gcd.a; b/ for all k > 0.
b) .d j a AND d j b/ IFF d j gcd.a; b/.
c) If gcd.a; b/ D 1 and gcd.a; c/ D 1, then gcd.a; bc/ D 1.
d) If a j bc and gcd.a; b/ D 1, then a j c.
Showing how all these facts follow from Theorem 9.2.2 that gcd is a linear combination
is a good exercise (Problem 9.11).
These properties are also simple consequences of the fact that integers factor
into primes in a unique way (Theorem 9.4.1). But we’ll need some of these facts
to prove unique factorization in Section 9.4, so proving them by appeal to unique
factorization would be circular.
9.3 Prime Mysteries
Some of the greatest mysteries and insights in number theory concern properties of
prime numbers:
Definition 9.3.1. A prime is a number greater than 1 that is divisible only by itself
and 1. A number other than 0, 1, and 1 that is not a prime is called composite. 7
Here are three famous mysteries:
Twin Prime Conjecture There are infinitely many primes p such that p C 2 is also
a prime.
In 1966, Chen showed that there are infinitely many primes p such that p C2
is the product of at most two primes. So the conjecture is known to be almost
true!
Conjectured Inefficiency of Factoring Given the product of two large primes n D
pq, there is no efficient procedure to recover the primes p and q. That is,
no polynomial time procedure (see Section 3.5) is guaranteed to find p and
7 So 0, 1, and 1 are the only integers that are neither prime nor composite.