Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 306 — #314
306
Chapter 9
Number Theory
Figure 9.1
Alan Turing
Since n=q 1 < n, this can’t be true, so we conclude that p 1 < q 1 .
Since the p i ’s are weakly decreasing, all the p i ’s are less than q 1 . But
q 1 j n D p 1 p 2 p j ;
so Lemma 9.4.3 implies that q 1 divides one of the p i ’s, which contradicts the fact
that q 1 is bigger than all them.
9.5 Alan Turing
The man pictured in Figure 9.1 is Alan Turing, the most important figure in the
history of computer science. For decades, his fascinating life story was shrouded
by government secrecy, societal taboo, and even his own deceptions.
At age 24, Turing wrote a paper entitled On Computable Numbers, with an Application
to the Entscheidungsproblem. The crux of the paper was an elegant way
to model a computer in mathematical terms. This was a breakthrough, because it
allowed the tools of mathematics to be brought to bear on questions of computation.
For example, with his model in hand, Turing immediately proved that there exist
problems that no computer can solve—no matter how ingenious the programmer.
Turing’s paper is all the more remarkable because he wrote it in 1936, a full decade