Euclid's proof of the infinitude of primes

Euclid's proof of the infinitude of primes is a classic and well-known proof by the Greek mathematician Euclid that there are infinitely many prime numbers.


We proceed by contradiction. Suppose there are in fact only finitely many prime numbers, $p_1, p_2, p_3, \ldots, p_n$. Let $N = p_1 \cdot p_2 \cdot p_3 \cdots p_n + 1$. Since $N$ leaves a remainder of 1 when divided by any of our prime numbers $p_k$, it is not divisible by any of them. However, the Fundamental Theorem of Arithmetic states that all positive integers have a unique prime factorization. Therefore, $N$ must have a prime factor (possibly itself) that is not among our set of primes, $\{p_1, p_2, p_3, \ldots, p_n\}$. This means that $\{p_1, p_2, p_3, \ldots, p_n \}$ does not contain all prime numbers, which contradicts our original assumption. Therefore, there must be infinitely many primes.

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