Page 172 - HOW TO PROVE IT: A Structured Approach, Second Edition

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158 Proofs
the method used in the rest of the proof. Although the cases are not explicitly
labeled as cases in the proof, it is important to realize that the rest of the proof
has the form of a proof by cases. In case 1 we assume that m is prime, and in
case 2 we assume that it is a product of primes. In both cases we are able to
produce an unlisted prime as required to complete the proof.

Theorem 3.7.3. For every positive integer n, there is a sequence of n consec-
utive positive integers containing no primes.

Proof. Suppose n is a positive integer. Let x = (n + 1)! + 2. We will show that
none of the numbers x, x + 1, x + 2, ··· , x + (n − 1) is prime. Since this is a
sequence of n consecutive positive integers, this will prove the theorem.
To see that x is not prime, note that
x = 1 · 2 · 3 · 4 ··· (n + 1) + 2
= 2 · (1 · 3 · 4 ··· (n + 1) + 1).

Thus, x can be written as a product of two smaller positive integers, so x is not
prime.
Similarly, we have
x + 1 = 1 · 2 · 3 · 4 ··· (n + 1) + 3
= 3 · (1 · 2 · 4 ··· (n + 1) + 1),

so x + 1 is also not prime. In general, consider any number x + i, where
0 ≤ i ≤ n − 1. Then we have
x + i = 1 · 2 · 3 · 4 ··· (n + 1) + (i + 2)
= (i + 2) · (1 · 2 · 3 ··· (i + 1) · (i + 3) ··· (n + 1) + 1),
so x + i is not prime.

Commentary. A sequence of n consecutive positive integers is a sequence of
the form x, x + 1, x + 2,..., x + (n − 1), where x is a positive integer. Thus,
the logical form of the statement to be proven is ∀n > 0∃x > 0∀i(0 ≤ i ≤
n − 1 → x + i is not prime), where all variables range over the integers. The
overall plan of the proof is exactly what one would expect for a proof of a
statement of this form: We let n > 0 be arbitrary, specify a value for x, let i
be arbitrary, and then assume that 0 ≤ i ≤ n − 1 and prove that x + i is not
prime. As in the proof of Theorem 3.7.1, to prove that x + i is not prime we
show how to write it as a product of two smaller integers.
Before the demonstration that x + i is not prime, where i is an arbitrary
integer between 0 and n − 1, the proof includes verifications that x and x + 1