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
There is a proof, and it takes advantage of the fact that a number is either prime, or a result of multiplying primes together. One of those two statements is always true.
So…
If there *was* a last prime, then you can count them and collect them. With that collection, multiply them ALL together.. 2 * 3 * 5 * 7 * 11 * 13 * and so on and so forth up to that last prime. Now take this result and then add 1.
What is this new number? It can’t be prime, it’s obviously *way* bigger than the biggest prime that exists after all. It also can’t be composite because that +1 on the end made it not divisible by any prime number that exists since we just used *all of them*. But it’s also a valid number in that all I’ve done is multiply a bunch of times (but not infinitely many) and then added 1. That’s normal math.
This number we invented cannot exist because it violates other established rules. We call this a contradiction because we’ve said that something is both true and false at the same time (in this case, the true statement is from what I said in the first paragraph). Therefore we have made a mistake in our assumptions, and the last assumption we made was wrong. Flip it around. Our last (and only) assumption was that there was such a thing as a last prime, so that is wrong. Therefore, there are infinite primes.
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