How do we know there are an infinite number of prime numbers? How do we know that once you get to a certain point, they don’t all end up as composite numbers?
In: Mathematics
Good question, and one with a relatively straightforward answet/proof! Let’s assume that there is a “biggest prime number.” That means I could create a composite number by multiplying ALL the prime numbers together (if there’s a biggest one, then the number of primes is finite).
This number is kind of the MOST composite number. It’s divisible by EVERY prime number by definition. So… what happens if we add 1 to it?
Now, the remainder when we divide by a prime number will be 1. That makes this “big number + 1” unable to be divided by any prime, and that means it must be prime itself… but this prime is bigger than the biggest prime number (thanks to the multiplication), which makes this a logical contradiction.
Contradictions don’t exist in well-defined systems, so the issue must be our initial assumption: that there actually is a “biggest prime number.” Hence the primes go on forever because if they didn’t, we’d have an impossible situation on our hands!
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