gen-briefly-explain-the-infinitude-of-primes-and-one-common-misc-86eb

Briefly explain the infinitude of primes and one common misconception about it.

# The Infinitude of Primes **The basic idea:** There are infinitely many prime numbers—no matter how far you count, there's always another prime waiting to be found. **Euclid's elegant proof (circa 300 BCE):** Suppose there were only finitely many primes. Multiply them all together and add 1. This new number isn't divisible by any of the primes on our "complete" list (it leaves remainder 1 when divided by each). So either this number is prime itself, or it has a prime factor we missed. Either way, our list wasn't complete—contradiction! ## Common Misconception Many people misunderstand Euclid's proof to claim that **multiplying all known primes and adding 1 always produces a new prime**. This is false! For example: 2 × 3 × 5 × 7 × 11 × 13 + 1 = 30,031 = 59 × 509 (composite, not prime) The proof only shows this number must have a prime factor *not on the original list*—it doesn't need to be prime itself. The distinction is subtle but important.