Are prime numbers infinite?

Yes, prime numbers are infinite.

Proof of the Infinitude of Prime Numbers (Euclid's Proof)

Euclid's proof, dating back over 2300 years, elegantly demonstrates that there is no largest prime number. The proof proceeds by contradiction:

1. Assumption: Let's assume that there are only finitely many prime numbers. We can list them all: p₁, p₂, p₃, ..., p_n.

2. Construction: Now, construct a new number, N, by multiplying all the primes in our list together and adding 1: N = (p₁ p₂ p₃ ... p_n) + 1.

3. Analysis: Consider the number N. There are two possibilities:

   • Case 1: N is prime. If N is prime, then we've found a prime number that is not in our original list (because N is larger than any prime in the list). This contradicts our initial assumption that we had a complete list of all prime numbers.

   • Case 2: N is composite. If N is composite (not prime), it must be divisible by at least one prime number. Let's call that prime number 'p'.

     • Could 'p' be one of the primes in our original list (p₁, p₂, p₃, ..., p_n)? No, it cannot. Because if 'p' were in the list, then 'p' would divide the product (p₁ p₂ p₃ ... p_n). Therefore, 'p' would have to divide N minus (p₁ p₂ p₃ ... p_n), which equals 1. But no prime number can divide 1 (except itself, which leads to p = 1, and 1 is not prime). This is a contradiction.

     • Therefore, 'p' must be a prime number not in our original list. Again, this contradicts our initial assumption.

4. Conclusion: In both possible cases (N is prime or N is composite), we arrive at a contradiction. Therefore, our initial assumption that there are finitely many prime numbers must be false. Consequently, there must be infinitely many prime numbers.

In simpler terms, no matter how many prime numbers you find, you can always find another one. This fundamental concept has profound implications in number theory and cryptography.