Concrete mathematics : a foundation for computer science

108 NUMBER THEORY
only finitely many primes, say k of them--, 3, 5, . . . , Pk. Then, said Euclid,
we should consider the number
M = 2’3’5’..:Pk + 1.
None of the k primes can divide M, because each divides M - 1. Thus there
must be some other prime that divides M; perhaps M itself is prime. This
contradicts our assumption that 2, 3, . . . , Pk are the only primes, so there
must indeed be infinitely many.
Euclid’s proof suggests that we define Euclid numbers by the recurrence
en = elez...e,-1 + 1, whenn>l. (4.16)
The sequence starts out
el =I+1 =2;
e2 =2+1 =3;
e3 = 2.3+1 = 7;
e4 = 2.3.7+1 = 43;
these are all prime. But the next case, e5, is 1807 = 13.139. It turns out that
e6 = 3263443 is prime, while
e7 = 547.607.1033.31051;
e8 = 29881~67003~9119521~6212157481.
It is known that es, . . . , e17 are composite, and the remaining e, are probably
composite as well. However, the Euclid numbers are all reZatiweZy prime to
each other; that is,
gcd(e,,e,) = 1 , when m # n.
Euclid’s algorithm (what else?) tells us this in three short steps, because
e, mod e, = 1 when n > m:
gc4em,e,) = gcd(l,e,) = gcd(O,l) = 1 ,
Therefore, if we let qj be the smallest factor of ej for all j 3 1, the primes 41,
q2, (73, . . . are all different. This is a sequence of infinitely many primes.
Let’s pause to consider the Euclid numbers from the standpoint of Chapter
1. Can we express e, in closed form? Recurrence (4.16) can be simplified
by removing the three dots: If n > 1 we have
en = el . . . en-2en-l + 1 = (en-l -l)e,-j fl = &,-qp, + 1.
cdot 7rpLjro1
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&i murb~ 706
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-Euclid [SO]
[Translation:
“There are more
primes than in
any given set
of primes. “1