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88
5. NUMBER THEORY
How many prime numbers are there?
Theorem. There is an infinite number of primes.
Proof. We us indirect reasoning.
Suppose there is a finite number of primes, say
2, 3, 5, 7, 11, . . . , p,
where p is the greatest prime. Let
N = 2 · 3 · 5 · 7 · 11 · · · p + 1.
N > 1 and N is greater than any prime.
But when N is divided by any prime, the remainder is always 1.
So N is not 1, is not prime, and is not a composite, which is impossible.
Thus there must be infinitely many primes.
⇤
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