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Number Theory · Axiom Academy
Do prime numbers go on forever, or does the sequence eventually end? First, let's recall what we know about prime numbers. Use the slider to explore the first several primes. Imagine we're living in ancient Greece, around 300 BCE. Euclid asks: "What if there were only finitely many primes?" Suppose this is the complete list of ALL primes: What if we assumed this list contained EVERY prime that exists? What could we do with this "complete" list? Let's build this special number. If 2, 3, 5, 7, and 11 were ALL the primes, we would calculate: Now we've reached the heart of Euclid's proof. Our number 2311 creates a logical puzzle... • It's not divisible by 2 (leaves remainder 1) • It's not divisible by 3 (leaves remainder 1) • It's not divisible by 5 (leaves remainder 1) • It's not divisible by 7 (leaves remainder 1) • It's not divisible by 11 (leaves remainder 1) But every number is either prime or divisible by a prime! Click below to explore the two possibilities... What can we conclude about 2311? Assume there are only finitely many primes. Multiply them all together and add 1. This new number isn't divisible by any prime on the list—which means either it's a new prime, or it has a prime factor we didn't know about. Either way, our list wasn't complete! This is called proof by contradiction . We assumed the opposite of what we wanted to prove, then showed this leads to an impossible situation. Therefore, our assumption must be false!
This is the written version of the interactive lesson above. See the full Number Theory course.