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Chapter 10: Problem 54

Prove that there are infinitely many prime numbers.

Short Answer

Expert verified
We arrive at a contradiction when we assume that there are only finitely many prime numbers, hence the proof concludes that there must be infinitely many prime numbers.

Step by step solution

01

Assume the opposite

We start by assuming the opposite: there are only finitely many prime numbers. Call these primes \( p_1, p_2, ..., p_n \) and let \( P \) be the product of these primes plus 1, i.e., \( P = p_1p_2...p_n + 1 \).
02

Prime Divisor

Every number greater than 1 has a prime divisor. Since \( P \) is greater than any of our list of primes \( p_i \), it is not in our list. We denote a prime divisor of \( P \) as \( p \).
03

Contradiction

There exists a contradiction here. If \( p \) is in our original list of primes, then it will not divide \( P \) perfectly as there will be a remainder of 1; on the other hand, if \( p \) is not in our original list, it contradicts our assumption that \( p_1, p_2, ..., p_n \) are all the prime numbers. Thus, we can conclude that there are indeed infinitely many prime numbers.

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